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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<a href="../../chapter_computational_complexity/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 2 章 複雜度分析
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 2 章 複雜度分析
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<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
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<span class="md-ellipsis">
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2.1 演算法效率評估
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</span>
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<span class="md-ellipsis">
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2.2 迭代與遞迴
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</span>
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<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
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<span class="md-ellipsis">
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2.3 時間複雜度
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
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<span class="md-ellipsis">
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2.4 空間複雜度
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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2.5 小結
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</a>
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<li class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
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第 3 章 資料結構
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</span>
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</a>
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<label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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第 3 章 資料結構
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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<span class="md-ellipsis">
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3.1 資料結構分類
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<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
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<span class="md-ellipsis">
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3.2 基本資料型別
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<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.3 數字編碼 *
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<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.4 字元編碼 *
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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3.5 小結
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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_6" >
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<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 4 章 陣列與鏈結串列
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</span>
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</a>
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<label class="md-nav__link " for="__nav_6" id="__nav_6_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_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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第 4 章 陣列與鏈結串列
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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_array_and_linkedlist/array/" class="md-nav__link">
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<span class="md-ellipsis">
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4.1 陣列
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</span>
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 鏈結串列
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3 串列
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4 記憶體與快取 *
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</span>
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<span class="md-status md-status--new" title="最近新增">
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.5 小結
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</span>
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</a>
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</li>
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</ul>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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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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第 5 章 堆疊與佇列
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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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第 5 章 堆疊與佇列
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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 堆疊
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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_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 佇列
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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_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 雙向佇列
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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_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4 小結
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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--active md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
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<div class="md-nav__link md-nav__container">
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<a href="../" 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">
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第 6 章 雜湊表
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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="true">
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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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第 6 章 雜湊表
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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="../hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
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6.1 雜湊表
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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="../hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2 雜湊衝突
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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="../hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3 雜湊演算法
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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--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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<span class="md-ellipsis">
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6.4 小結
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<a href="./" class="md-nav__link md-nav__link--active">
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<span class="md-ellipsis">
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6.4 小結
|
|
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目錄">
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
|
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目錄
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#1" class="md-nav__link">
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<span class="md-ellipsis">
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1. 重點回顧
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|
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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="#2-q-a" class="md-nav__link">
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<span class="md-ellipsis">
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2. Q & A
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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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</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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第 7 章 樹
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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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第 7 章 樹
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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 二元樹
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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 二元樹走訪
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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 二元樹陣列表示
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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 二元搜尋樹
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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 *
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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/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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7.6 小結
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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_10" >
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<div class="md-nav__link md-nav__container">
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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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第 8 章 堆積
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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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</label>
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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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第 8 章 堆積
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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_heap/heap/" class="md-nav__link">
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<span class="md-ellipsis">
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8.1 堆積
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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_heap/build_heap/" class="md-nav__link">
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<span class="md-ellipsis">
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8.2 建堆積操作
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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_heap/top_k/" class="md-nav__link">
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<span class="md-ellipsis">
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8.3 Top-k 問題
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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 小結
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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_11" >
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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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第 9 章 圖
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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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<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_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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第 9 章 圖
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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_graph/graph/" class="md-nav__link">
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<span class="md-ellipsis">
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9.1 圖
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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_graph/graph_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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9.2 圖基礎操作
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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_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3 圖的走訪
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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_graph/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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9.4 小結
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</span>
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</a>
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</li>
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</ul>
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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_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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第 10 章 搜尋
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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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</label>
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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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第 10 章 搜尋
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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 二分搜尋
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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 二分搜尋插入點
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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 二分搜尋邊界
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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/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 雜湊最佳化策略
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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/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 重識搜尋演算法
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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 小結
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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_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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第 11 章 排序
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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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</label>
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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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第 11 章 排序
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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_sorting/sorting_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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11.1 排序演算法
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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 選擇排序
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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 泡沫排序
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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 插入排序
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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/quick_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.5 快速排序
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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 合併排序
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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/heap_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.7 堆積排序
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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 桶排序
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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 計數排序
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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 基數排序
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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 小結
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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_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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第 12 章 分治
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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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</label>
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</div>
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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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|
第 12 章 分治
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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_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
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<span class="md-ellipsis">
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12.1 分治演算法
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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/binary_search_recur/" class="md-nav__link">
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<span class="md-ellipsis">
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12.2 分治搜尋策略
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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 構建樹問題
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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 河內塔問題
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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 小結
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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_15" >
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<div class="md-nav__link md-nav__container">
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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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第 13 章 回溯
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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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</label>
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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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|
第 13 章 回溯
|
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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_backtracking/backtracking_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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13.1 回溯演算法
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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_backtracking/permutations_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.2 全排列問題
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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_backtracking/subset_sum_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.3 子集和問題
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.4 N 皇后問題
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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13.5 小結
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</a>
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<li class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
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第 14 章 動態規劃
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<span class="md-nav__icon md-icon"></span>
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第 14 章 動態規劃
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
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<span class="md-ellipsis">
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14.1 初探動態規劃
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<a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
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<span class="md-ellipsis">
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14.2 DP 問題特性
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<a href="../../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
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<span class="md-ellipsis">
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14.3 DP 解題思路
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<span class="md-ellipsis">
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14.4 0-1 背包問題
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<a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.5 完全背包問題
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<a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.6 編輯距離問題
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</span>
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<a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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14.7 小結
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</a>
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<span class="md-ellipsis">
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第 15 章 貪婪
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</span>
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</a>
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<span class="md-nav__icon md-icon"></span>
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第 15 章 貪婪
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_greedy/greedy_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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15.1 貪婪演算法
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</span>
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<a href="../../chapter_greedy/fractional_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.2 分數背包問題
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</span>
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</a>
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<a href="../../chapter_greedy/max_capacity_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.3 最大容量問題
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</span>
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<a href="../../chapter_greedy/max_product_cutting_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.4 最大切分乘積問題
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_greedy/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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15.5 小結
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</span>
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</a>
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<span class="md-ellipsis">
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第 16 章 附錄
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</span>
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</a>
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第 16 章 附錄
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<ul class="md-nav__list" data-md-scrollfix>
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16.1 程式設計環境安裝
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</span>
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</a>
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<span class="md-ellipsis">
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16.2 一起參與創作
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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16.3 術語表
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<h1 id="64">6.4 小結<a class="headerlink" href="#64" title="Permanent link">¶</a></h1>
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<h3 id="1">1. 重點回顧<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
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<ul>
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<li>輸入 <code>key</code> ，雜湊表能夠在 <span class="arithmatex">\(O(1)\)</span> 時間內查詢到 <code>value</code> ，效率非常高。</li>
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<li>常見的雜湊表操作包括查詢、新增鍵值對、刪除鍵值對和走訪雜湊表等。</li>
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<li>雜湊函式將 <code>key</code> 對映為陣列索引，從而訪問對應桶並獲取 <code>value</code> 。</li>
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<li>兩個不同的 <code>key</code> 可能在經過雜湊函式後得到相同的陣列索引，導致查詢結果出錯，這種現象被稱為雜湊衝突。</li>
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<li>雜湊表容量越大，雜湊衝突的機率就越低。因此可以透過擴容雜湊表來緩解雜湊衝突。與陣列擴容類似，雜湊表擴容操作的開銷很大。</li>
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<li>負載因子定義為雜湊表中元素數量除以桶數量，反映了雜湊衝突的嚴重程度，常用作觸發雜湊表擴容的條件。</li>
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<li>鏈式位址透過將單個元素轉化為鏈結串列，將所有衝突元素儲存在同一個鏈結串列中。然而，鏈結串列過長會降低查詢效率，可以透過進一步將鏈結串列轉換為紅黑樹來提高效率。</li>
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<li>開放定址透過多次探測來處理雜湊衝突。線性探查使用固定步長，缺點是不能刪除元素，且容易產生聚集。多次雜湊使用多個雜湊函式進行探測，相較線性探查更不易產生聚集，但多個雜湊函式增加了計算量。</li>
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<li>不同程式語言採取了不同的雜湊表實現。例如，Java 的 <code>HashMap</code> 使用鏈式位址，而 Python 的 <code>Dict</code> 採用開放定址。</li>
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<li>在雜湊表中，我們希望雜湊演算法具有確定性、高效率和均勻分佈的特點。在密碼學中，雜湊演算法還應該具備抗碰撞性和雪崩效應。</li>
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<li>雜湊演算法通常採用大質數作為模數，以最大化地保證雜湊值均勻分佈，減少雜湊衝突。</li>
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<li>常見的雜湊演算法包括 MD5、SHA-1、SHA-2 和 SHA-3 等。MD5 常用於校驗檔案完整性，SHA-2 常用於安全應用與協議。</li>
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<li>程式語言通常會為資料型別提供內建雜湊演算法，用於計算雜湊表中的桶索引。通常情況下，只有不可變物件是可雜湊的。</li>
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</ul>
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<h3 id="2-q-a">2. Q & A<a class="headerlink" href="#2-q-a" title="Permanent link">¶</a></h3>
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<p><strong>Q</strong>：雜湊表的時間複雜度在什麼情況下是 <span class="arithmatex">\(O(n)\)</span> ？</p>
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<p>當雜湊衝突比較嚴重時，雜湊表的時間複雜度會退化至 <span class="arithmatex">\(O(n)\)</span> 。當雜湊函式設計得比較好、容量設定比較合理、衝突比較平均時，時間複雜度是 <span class="arithmatex">\(O(1)\)</span> 。我們使用程式語言內建的雜湊表時，通常認為時間複雜度是 <span class="arithmatex">\(O(1)\)</span> 。</p>
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<p><strong>Q</strong>：為什麼不使用雜湊函式 <span class="arithmatex">\(f(x) = x\)</span> 呢？這樣就不會有衝突了。</p>
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<p>在 <span class="arithmatex">\(f(x) = x\)</span> 雜湊函式下，每個元素對應唯一的桶索引，這與陣列等價。然而，輸入空間通常遠大於輸出空間（陣列長度），因此雜湊函式的最後一步往往是對陣列長度取模。換句話說，雜湊表的目標是將一個較大的狀態空間對映到一個較小的空間，並提供 <span class="arithmatex">\(O(1)\)</span> 的查詢效率。</p>
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<p><strong>Q</strong>：雜湊表底層實現是陣列、鏈結串列、二元樹，但為什麼效率可以比它們更高呢？</p>
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<p>首先，雜湊表的時間效率變高，但空間效率變低了。雜湊表有相當一部分記憶體未使用。</p>
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<p>其次，只是在特定使用場景下時間效率變高了。如果一個功能能夠在相同的時間複雜度下使用陣列或鏈結串列實現，那麼通常比雜湊表更快。這是因為雜湊函式計算需要開銷，時間複雜度的常數項更大。</p>
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<p>最後，雜湊表的時間複雜度可能發生劣化。例如在鏈式位址中，我們採取在鏈結串列或紅黑樹中執行查詢操作，仍然有退化至 <span class="arithmatex">\(O(n)\)</span> 時間的風險。</p>
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<p><strong>Q</strong>：多次雜湊有不能直接刪除元素的缺陷嗎？標記為已刪除的空間還能再次使用嗎？</p>
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<p>多次雜湊是開放定址的一種，開放定址法都有不能直接刪除元素的缺陷，需要透過標記刪除。標記為已刪除的空間可以再次使用。當將新元素插入雜湊表，並且透過雜湊函式找到標記為已刪除的位置時，該位置可以被新元素使用。這樣做既能保持雜湊表的探測序列不變，又能保證雜湊表的空間使用率。</p>
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<p><strong>Q</strong>：為什麼線上性探查中，查詢元素的時候會出現雜湊衝突呢？</p>
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<p>查詢的時候透過雜湊函式找到對應的桶和鍵值對，發現 <code>key</code> 不匹配，這就代表有雜湊衝突。因此，線性探查法會根據預先設定的步長依次向下查詢，直至找到正確的鍵值對或無法找到跳出為止。</p>
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<p><strong>Q</strong>：為什麼雜湊表擴容能夠緩解雜湊衝突？</p>
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<p>雜湊函式的最後一步往往是對陣列長度 <span class="arithmatex">\(n\)</span> 取模（取餘），讓輸出值落在陣列索引範圍內；在擴容後，陣列長度 <span class="arithmatex">\(n\)</span> 發生變化，而 <code>key</code> 對應的索引也可能發生變化。原先落在同一個桶的多個 <code>key</code> ，在擴容後可能會被分配到多個桶中，從而實現雜湊衝突的緩解。</p>
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