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<li class="md-nav__item">
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<a href="../linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 Linked list
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3 List
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../ram_and_cache/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4 Memory and cache *
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.5 Summary
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
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<span class="md-ellipsis">
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Chapter 5. Stack and queue
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</span>
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</a>
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<label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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Chapter 5. Stack and queue
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1 Stack
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 Queue
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 Double-ended queue
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
|
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5.4 Summary
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|
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_hashing/" class="md-nav__link ">
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|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
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<span class="md-ellipsis">
|
|
|
Chapter 6. Hash table
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</span>
|
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</a>
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<label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
|
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Chapter 6. Hash table
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
|
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|
6.1 Hash table
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
|
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6.2 Hash collision
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
|
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6.3 Hash algorithm
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
|
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6.4 Summary
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_tree/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
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<span class="md-ellipsis">
|
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|
Chapter 7. Tree
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</span>
|
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</a>
|
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|
<label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
|
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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|
|
<label class="md-nav__title" for="__nav_9">
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|
<span class="md-nav__icon md-icon"></span>
|
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Chapter 7. Tree
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.1 Binary tree
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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7.2 Binary tree traversal
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.3 Array Representation of tree
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.4 Binary Search tree
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</span>
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.5 AVL tree *
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</a>
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<span class="md-ellipsis">
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7.6 Summary
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</a>
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<a href="../../chapter_heap/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
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<span class="md-ellipsis">
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Chapter 8. Heap
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</span>
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</a>
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<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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Chapter 8. Heap
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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8.1 Heap
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</span>
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</a>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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8.2 Building a heap
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</span>
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</a>
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
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<span class="md-ellipsis">
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8.3 Top-k problem
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<li class="md-nav__item">
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<a href="../../chapter_heap/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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8.4 Summary
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</span>
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</a>
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_graph/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
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<span class="md-ellipsis">
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Chapter 9. Graph
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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Chapter 9. Graph
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_graph/graph/" class="md-nav__link">
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<span class="md-ellipsis">
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9.1 Graph
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</span>
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</a>
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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9.2 Basic graph operations
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</span>
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</a>
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3 Graph traversal
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</span>
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</a>
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<span class="md-ellipsis">
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9.4 Summary
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</span>
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</a>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_searching/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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Chapter 10. Searching
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</span>
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</a>
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<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
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Chapter 10. Searching
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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<span class="md-ellipsis">
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10.1 Binary search
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 Binary search insertion
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<span class="md-ellipsis">
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10.3 Binary search boundaries
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 Hashing optimization strategies
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 Search algorithms revisited
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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10.6 Summary
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_sorting/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
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<span class="md-ellipsis">
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Chapter 11. Sorting
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</span>
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</a>
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<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_13">
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<span class="md-nav__icon md-icon"></span>
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Chapter 11. Sorting
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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11.1 Sorting algorithms
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.2 Selection sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.3 Bubble sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.4 Insertion sort
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.5 Quick sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.6 Merge sort
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.7 Heap sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.8 Bucket sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.9 Counting sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.10 Radix sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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11.11 Summary
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
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<span class="md-ellipsis">
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Chapter 12. Divide and conquer
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</span>
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</a>
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<label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
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Chapter 12. Divide and conquer
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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12.1 Divide and conquer algorithms
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
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<span class="md-ellipsis">
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12.2 Divide and conquer search strategy
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.3 Building binary tree problem
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.4 Tower of Hanoi Problem
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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12.5 Summary
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<a href="../../chapter_backtracking/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
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<span class="md-ellipsis">
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Chapter 13. Backtracking
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</span>
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</a>
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<label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_15">
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<span class="md-nav__icon md-icon"></span>
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Chapter 13. Backtracking
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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13.1 Backtracking algorithms
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<span class="md-ellipsis">
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13.2 Permutation problem
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<span class="md-ellipsis">
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13.3 Subset sum problem
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13.4 n queens problem
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13.5 Summary
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14.1 Introduction to dynamic programming
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14.2 Characteristics of DP problems
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14.3 DP problem-solving approach¶
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14.4 0-1 Knapsack problem
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14.5 Unbounded knapsack problem
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14.6 Edit distance problem
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Chapter 15. Greedy
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15.1 Greedy algorithms
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<h1 id="41-array">4.1 Array<a class="headerlink" href="#41-array" title="Permanent link">¶</a></h1>
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<p>An <u>array</u> is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an <u>index</u>. Please refer to Figure 4-1 to observe how arrays work and grasp these key terms.</p>
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<p><a class="glightbox" href="../array.assets/array_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array definition and storage method" class="animation-figure" src="../array.assets/array_definition.png" /></a></p>
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<p align="center"> Figure 4-1 Array definition and storage method </p>
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<h2 id="411-common-operations-on-arrays">4.1.1 Common operations on arrays<a class="headerlink" href="#411-common-operations-on-arrays" title="Permanent link">¶</a></h2>
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<h3 id="1-initializing-arrays">1. Initializing arrays<a class="headerlink" href="#1-initializing-arrays" title="Permanent link">¶</a></h3>
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<p>Arrays can be initialized in two ways depending on the needs: either without initial values or with specified initial values. When initial values are not specified, most programming languages will set the array elements to <span class="arithmatex">\(0\)</span>:</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Zig</label></div>
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize array</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="n">arr</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span> <span class="c1"># [ 0, 0, 0, 0, 0 ]</span>
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Stored on stack</span>
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<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kt">int</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
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<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
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<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1">// Stored on heap (manual memory release needed)</span>
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<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="kt">int</span><span class="o">*</span><span class="w"> </span><span class="n">arr1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
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<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="kt">int</span><span class="o">*</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize array */</span>
|
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="mi">5</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// { 0, 0, 0, 0, 0 }</span>
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="m">5</span><span class="p">];</span><span class="w"> </span><span class="c1">// [ 0, 0, 0, 0, 0 ]</span>
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<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">];</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">var</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="kt">int</span>
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<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="c1">// In Go, specifying the length ([5]int) denotes an array, while not specifying it ([]int) denotes a slice.</span>
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<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="c1">// Since Go's arrays are designed to have compile-time fixed length, only constants can be used to specify the length.</span>
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<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// For convenience in implementing the extend() method, the Slice will be considered as an Array here.</span>
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<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="nx">nums</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">let</span> <span class="nv">arr</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="mi">5</span><span class="p">)</span> <span class="c1">// [0, 0, 0, 0, 0]</span>
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<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">let</span> <span class="nv">nums</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">var</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">5</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
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<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">var</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">5</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">];</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">let</span><span class="w"> </span><span class="nx">arr</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">5</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
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<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">5</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">];</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// [0, 0, 0, 0, 0]</span>
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<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">];</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kd">let</span><span class="w"> </span><span class="n">arr</span>: <span class="nb">Vec</span><span class="o"><</span><span class="kt">i32</span><span class="o">></span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">];</span><span class="w"> </span><span class="c1">// [0, 0, 0, 0, 0]</span>
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<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">nums</span>: <span class="nb">Vec</span><span class="o"><</span><span class="kt">i32</span><span class="o">></span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">];</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Initialize array */</span>
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<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">};</span><span class="w"> </span><span class="c1">// { 0, 0, 0, 0, 0 }</span>
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">// Initialize array</span>
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<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="kr">var</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="n">_</span><span class="p">]</span><span class="kt">i32</span><span class="p">{</span><span class="mi">0</span><span class="p">}</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="mi">5</span><span class="p">;</span><span class="w"> </span><span class="c1">// { 0, 0, 0, 0, 0 }</span>
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<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="kr">var</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="n">_</span><span class="p">]</span><span class="kt">i32</span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h3 id="2-accessing-elements">2. Accessing elements<a class="headerlink" href="#2-accessing-elements" title="Permanent link">¶</a></h3>
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<p>Elements in an array are stored in contiguous memory spaces, making it simpler to compute each element's memory address. The formula shown in the Figure below aids in determining an element's memory address, utilizing the array's memory address (specifically, the first element's address) and the element's index. This computation streamlines direct access to the desired element.</p>
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<p><a class="glightbox" href="../array.assets/array_memory_location_calculation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memory address calculation for array elements" class="animation-figure" src="../array.assets/array_memory_location_calculation.png" /></a></p>
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<p align="center"> Figure 4-2 Memory address calculation for array elements </p>
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<p>As observed in the above illustration, array indexing conventionally begins at <span class="arithmatex">\(0\)</span>. While this might appear counterintuitive, considering counting usually starts at <span class="arithmatex">\(1\)</span>, within the address calculation formula, <strong>an index is essentially an offset from the memory address</strong>. For the first element's address, this offset is <span class="arithmatex">\(0\)</span>, validating its index as <span class="arithmatex">\(0\)</span>.</p>
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<p>Accessing elements in an array is highly efficient, allowing us to randomly access any element in <span class="arithmatex">\(O(1)\)</span> time.</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:14"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label><label for="__tabbed_2_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span> <span class="nf">random_access</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
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<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w"> </span><span class="sd">"""Random access to elements"""</span>
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<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a> <span class="c1"># Randomly select a number from the interval [0, len(nums)-1]</span>
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<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a> <span class="n">random_index</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a> <span class="c1"># Retrieve and return a random element</span>
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<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a> <span class="n">random_num</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">random_index</span><span class="p">]</span>
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<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a> <span class="k">return</span> <span class="n">random_num</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Random access to elements */</span>
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<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">randomAccess</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w"> </span><span class="c1">// Randomly select a number in the interval [0, nums.length)</span>
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<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ThreadLocalRandom</span><span class="p">.</span><span class="na">current</span><span class="p">().</span><span class="na">nextInt</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">);</span>
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<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w"> </span><span class="c1">// Retrieve and return a random element</span>
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<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">randomIndex</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span><span class="p">;</span>
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<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">array</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">RandomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">random_access</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">random_access</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">randomAccess</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<h3 id="3-inserting-elements">3. Inserting elements<a class="headerlink" href="#3-inserting-elements" title="Permanent link">¶</a></h3>
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<p>Array elements are tightly packed in memory, with no space available to accommodate additional data between them. Illustrated in Figure below, inserting an element in the middle of an array requires shifting all subsequent elements back by one position to create room for the new element.</p>
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<p><a class="glightbox" href="../array.assets/array_insert_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array element insertion example" class="animation-figure" src="../array.assets/array_insert_element.png" /></a></p>
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<p align="center"> Figure 4-3 Array element insertion example </p>
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<p>It's important to note that due to the fixed length of an array, inserting an element will unavoidably result in the loss of the last element in the array. Solutions to address this issue will be explored in the "List" chapter.</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:14"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><input id="__tabbed_3_14" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label><label for="__tabbed_3_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="k">def</span> <span class="nf">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
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<a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="w"> </span><span class="sd">"""Insert element num at `index`"""</span>
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<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a> <span class="c1"># Move all elements after `index` one position backward</span>
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<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
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<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a> <span class="c1"># Assign num to the element at index</span>
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<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a> <span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Insert element num at `index` */</span>
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<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w"> </span><span class="c1">// Move all elements after `index` one position backward</span>
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<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w"> </span><span class="c1">// Assign num to the element at index</span>
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<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
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<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">array</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">insert</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="bp">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h3 id="4-deleting-elements">4. Deleting elements<a class="headerlink" href="#4-deleting-elements" title="Permanent link">¶</a></h3>
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<p>Similarly, as depicted in Figure 4-4, to delete an element at index <span class="arithmatex">\(i\)</span>, all elements following index <span class="arithmatex">\(i\)</span> must be moved forward by one position.</p>
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<p><a class="glightbox" href="../array.assets/array_remove_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Array element deletion example" class="animation-figure" src="../array.assets/array_remove_element.png" /></a></p>
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<p align="center"> Figure 4-4 Array element deletion example </p>
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<p>Please note that after deletion, the former last element becomes "meaningless," hence requiring no specific modification.</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label><label for="__tabbed_4_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="k">def</span> <span class="nf">remove</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
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<a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="w"> </span><span class="sd">"""Remove the element at `index`"""</span>
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<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a> <span class="c1"># Move all elements after `index` one position forward</span>
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<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Remove the element at `index` */</span>
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<a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w"> </span><span class="c1">// Move all elements after `index` one position forward</span>
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<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">array</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">removeItem</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<p>In summary, the insertion and deletion operations in arrays present the following disadvantages:</p>
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<ul>
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<li><strong>High time complexity</strong>: Both insertion and deletion in an array have an average time complexity of <span class="arithmatex">\(O(n)\)</span>, where <span class="arithmatex">\(n\)</span> is the length of the array.</li>
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<li><strong>Loss of elements</strong>: Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.</li>
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<li><strong>Waste of memory</strong>: Initializing a longer array and utilizing only the front part results in "meaningless" end elements during insertion, leading to some wasted memory space.</li>
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</ul>
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<h3 id="5-traversing-arrays">5. Traversing arrays<a class="headerlink" href="#5-traversing-arrays" title="Permanent link">¶</a></h3>
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<p>In most programming languages, we can traverse an array either by using indices or by directly iterating over each element:</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="5:14"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label><label for="__tabbed_5_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="k">def</span> <span class="nf">traverse</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
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<a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="w"> </span><span class="sd">"""Traverse array"""</span>
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<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
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<a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a> <span class="c1"># Traverse array by index</span>
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<a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
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<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a> <span class="n">count</span> <span class="o">+=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
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<a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a> <span class="c1"># Traverse array elements</span>
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<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a> <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
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<a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a> <span class="n">count</span> <span class="o">+=</span> <span class="n">num</span>
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<a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a> <span class="c1"># Traverse both data index and elements</span>
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<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">num</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">nums</span><span class="p">):</span>
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|
<a id="__codelineno-55-12" name="__codelineno-55-12" href="#__codelineno-55-12"></a> <span class="n">count</span> <span class="o">+=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
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<a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a> <span class="n">count</span> <span class="o">+=</span> <span class="n">num</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Traverse array */</span>
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<a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
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<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w"> </span><span class="c1">// Traverse array by index</span>
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<a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w"> </span><span class="c1">// Traverse array elements</span>
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<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
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<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">array</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">traverse</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h3 id="6-finding-elements">6. Finding elements<a class="headerlink" href="#6-finding-elements" title="Permanent link">¶</a></h3>
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<p>Locating a specific element within an array involves iterating through the array, checking each element to determine if it matches the desired value.</p>
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<p>Because arrays are linear data structures, this operation is commonly referred to as "linear search."</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="6:14"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><input id="__tabbed_6_14" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label><label for="__tabbed_6_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="k">def</span> <span class="nf">find</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
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<a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="w"> </span><span class="sd">"""Search for a specified element in the array"""</span>
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<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
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<a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
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<a id="__codelineno-69-5" name="__codelineno-69-5" href="#__codelineno-69-5"></a> <span class="k">return</span> <span class="n">i</span>
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<a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Search for a specified element in the array */</span>
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<a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
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<a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
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<a id="__codelineno-71-6" name="__codelineno-71-6" href="#__codelineno-71-6"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-71-7" name="__codelineno-71-7" href="#__codelineno-71-7"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
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<a id="__codelineno-71-8" name="__codelineno-71-8" href="#__codelineno-71-8"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">array</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="bp">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">find</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h3 id="7-expanding-arrays">7. Expanding arrays<a class="headerlink" href="#7-expanding-arrays" title="Permanent link">¶</a></h3>
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<p>In complex system environments, ensuring the availability of memory space after an array for safe capacity extension becomes challenging. Consequently, in most programming languages, <strong>the length of an array is immutable</strong>.</p>
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<p>To expand an array, it's necessary to create a larger array and then copy the elements from the original array. This operation has a time complexity of <span class="arithmatex">\(O(n)\)</span> and can be time-consuming for large arrays. The code are as follows:</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="7:14"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><input id="__tabbed_7_14" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label><label for="__tabbed_7_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="k">def</span> <span class="nf">extend</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">enlarge</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
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<a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="w"> </span><span class="sd">"""Extend array length"""</span>
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<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a> <span class="c1"># Initialize an extended length array</span>
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<a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">+</span> <span class="n">enlarge</span><span class="p">)</span>
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<a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a> <span class="c1"># Copy all elements from the original array to the new array</span>
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<a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
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<a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a> <span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
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<a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a> <span class="c1"># Return the new array after expansion</span>
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<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a> <span class="k">return</span> <span class="n">res</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Extend array length */</span>
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<a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w"> </span><span class="c1">// Initialize an extended length array</span>
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<a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w"> </span><span class="c1">// Copy all elements from the original array to the new array</span>
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<a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w"> </span><span class="c1">// Return the new array after expansion</span>
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<a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
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<a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">array</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="bp">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-91-1" name="__codelineno-91-1" href="#__codelineno-91-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="kp">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">array.zig</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">extend</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h2 id="412-advantages-and-limitations-of-arrays">4.1.2 Advantages and limitations of arrays<a class="headerlink" href="#412-advantages-and-limitations-of-arrays" title="Permanent link">¶</a></h2>
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<p>Arrays are stored in contiguous memory spaces and consist of elements of the same type. This approach provides substantial prior information that systems can leverage to optimize the efficiency of data structure operations.</p>
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<ul>
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<li><strong>High space efficiency</strong>: Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.</li>
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<li><strong>Support for random access</strong>: Arrays allow <span class="arithmatex">\(O(1)\)</span> time access to any element.</li>
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<li><strong>Cache locality</strong>: When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.</li>
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</ul>
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<p>However, continuous space storage is a double-edged sword, with the following limitations:</p>
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<ul>
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|
<li><strong>Low efficiency in insertion and deletion</strong>: As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.</li>
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<li><strong>Fixed length</strong>: The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.</li>
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<li><strong>Space wastage</strong>: If the allocated array size exceeds the what is necessary, the extra space is wasted.</li>
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</ul>
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<h2 id="413-typical-applications-of-arrays">4.1.3 Typical applications of arrays<a class="headerlink" href="#413-typical-applications-of-arrays" title="Permanent link">¶</a></h2>
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<p>Arrays are fundamental and widely used data structures. They find frequent application in various algorithms and serve in the implementation of complex data structures.</p>
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<ul>
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<li><strong>Random access</strong>: Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.</li>
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<li><strong>Sorting and searching</strong>: Arrays are the most commonly used data structure for sorting and searching algorithms. Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.</li>
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<li><strong>Lookup tables</strong>: Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.</li>
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<li><strong>Machine learning</strong>: Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.</li>
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<li><strong>Data structure implementation</strong>: Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li>
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</ul>
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