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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--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">
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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="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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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="../../chapter_hashing/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="../../chapter_hashing/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="../../chapter_hashing/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">
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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 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>
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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>
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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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</a>
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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_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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<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 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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</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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</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 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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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" 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="m19.31 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="true">
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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="../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="../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="../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="../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>
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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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10.5 重識搜尋演算法
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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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10.5 重識搜尋演算法
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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="#1051" class="md-nav__link">
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<span class="md-ellipsis">
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10.5.1 暴力搜尋
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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="#1052" class="md-nav__link">
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<span class="md-ellipsis">
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10.5.2 自適應搜尋
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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="#1053" class="md-nav__link">
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<span class="md-ellipsis">
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10.5.3 搜尋方法選取
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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">
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<a href="../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 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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</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 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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第 13 章 回溯
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13.1 回溯演算法
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13.2 全排列問題
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13.3 子集和問題
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13.4 N 皇后問題
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13.5 小結
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第 14 章 動態規劃
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第 14 章 動態規劃
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14.1 初探動態規劃
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14.2 DP 問題特性
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14.3 DP 解題思路
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14.4 0-1 背包問題
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<span class="md-ellipsis">
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14.5 完全背包問題
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<span class="md-ellipsis">
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14.6 編輯距離問題
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14.7 小結
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第 15 章 貪婪
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第 15 章 貪婪
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15.1 貪婪演算法
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15.2 分數背包問題
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15.3 最大容量問題
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15.4 最大切分乘積問題
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15.5 小結
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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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</div>
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16.3 術語表
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10.5.1 暴力搜尋
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10.5.3 搜尋方法選取
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href="https://github.com/krahets/hello-algo/tree/main/zh-hant/docs/chapter_searching/searching_algorithm_revisited.md"
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title="編輯此頁"
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M441 58.9 453.1 71c9.4 9.4 9.4 24.6 0 33.9L424 134.1 377.9 88 407 58.9c9.4-9.4 24.6-9.4 33.9 0zM209.8 256.2 344 121.9l46.1 46.1-134.3 134.2c-2.9 2.9-6.5 5-10.4 6.1L186.9 325l16.7-58.5c1.1-3.9 3.2-7.5 6.1-10.4zM373.1 25 175.8 222.2c-8.7 8.7-15 19.4-18.3 31.1l-28.6 100c-2.4 8.4-.1 17.4 6.1 23.6s15.2 8.5 23.6 6.1l100-28.6c11.8-3.4 22.5-9.7 31.1-18.3L487 138.9c28.1-28.1 28.1-73.7 0-101.8L474.9 25c-28.1-28.1-73.7-28.1-101.8 0zM88 64c-48.6 0-88 39.4-88 88v272c0 48.6 39.4 88 88 88h272c48.6 0 88-39.4 88-88V312c0-13.3-10.7-24-24-24s-24 10.7-24 24v112c0 22.1-17.9 40-40 40H88c-22.1 0-40-17.9-40-40V152c0-22.1 17.9-40 40-40h112c13.3 0 24-10.7 24-24s-10.7-24-24-24H88z"/></svg>
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<h1 id="105">10.5 重識搜尋演算法<a class="headerlink" href="#105" title="Permanent link">¶</a></h1>
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<p><u>搜尋演算法（searching algorithm）</u>用於在資料結構（例如陣列、鏈結串列、樹或圖）中搜索一個或一組滿足特定條件的元素。</p>
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<p>搜尋演算法可根據實現思路分為以下兩類。</p>
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<ul>
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<li><strong>透過走訪資料結構來定位目標元素</strong>，例如陣列、鏈結串列、樹和圖的走訪等。</li>
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<li><strong>利用資料組織結構或資料包含的先驗資訊，實現高效元素查詢</strong>，例如二分搜尋、雜湊查詢和二元搜尋樹查詢等。</li>
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</ul>
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<p>不難發現，這些知識點都已在前面的章節中介紹過，因此搜尋演算法對於我們來說並不陌生。在本節中，我們將從更加系統的視角切入，重新審視搜尋演算法。</p>
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<h2 id="1051">10.5.1 暴力搜尋<a class="headerlink" href="#1051" title="Permanent link">¶</a></h2>
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<p>暴力搜尋透過走訪資料結構的每個元素來定位目標元素。</p>
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<ul>
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<li>“線性搜尋”適用於陣列和鏈結串列等線性資料結構。它從資料結構的一端開始，逐個訪問元素，直到找到目標元素或到達另一端仍沒有找到目標元素為止。</li>
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<li>“廣度優先搜尋”和“深度優先搜尋”是圖和樹的兩種走訪策略。廣度優先搜尋從初始節點開始逐層搜尋，由近及遠地訪問各個節點。深度優先搜尋從初始節點開始，沿著一條路徑走到頭，再回溯並嘗試其他路徑，直到走訪完整個資料結構。</li>
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</ul>
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<p>暴力搜尋的優點是簡單且通用性好，<strong>無須對資料做預處理和藉助額外的資料結構</strong>。</p>
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<p>然而，<strong>此類演算法的時間複雜度為 <span class="arithmatex">\(O(n)\)</span></strong> ，其中 <span class="arithmatex">\(n\)</span> 為元素數量，因此在資料量較大的情況下效能較差。</p>
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<h2 id="1052">10.5.2 自適應搜尋<a class="headerlink" href="#1052" title="Permanent link">¶</a></h2>
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<p>自適應搜尋利用資料的特有屬性（例如有序性）來最佳化搜尋過程，從而更高效地定位目標元素。</p>
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<ul>
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<li>“二分搜尋”利用資料的有序性實現高效查詢，僅適用於陣列。</li>
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<li>“雜湊查詢”利用雜湊表將搜尋資料和目標資料建立為鍵值對對映，從而實現查詢操作。</li>
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<li>“樹查詢”在特定的樹結構（例如二元搜尋樹）中，基於比較節點值來快速排除節點，從而定位目標元素。</li>
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</ul>
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<p>此類演算法的優點是效率高，<strong>時間複雜度可達到 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span></strong> 。</p>
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<p>然而，<strong>使用這些演算法往往需要對資料進行預處理</strong>。例如，二分搜尋需要預先對陣列進行排序，雜湊查詢和樹查詢都需要藉助額外的資料結構，維護這些資料結構也需要額外的時間和空間開銷。</p>
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<div class="admonition tip">
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<p class="admonition-title">Tip</p>
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<p>自適應搜尋演算法常被稱為查詢演算法，<strong>主要用於在特定資料結構中快速檢索目標元素</strong>。</p>
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</div>
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<h2 id="1053">10.5.3 搜尋方法選取<a class="headerlink" href="#1053" title="Permanent link">¶</a></h2>
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<p>給定大小為 <span class="arithmatex">\(n\)</span> 的一組資料，我們可以使用線性搜尋、二分搜尋、樹查詢、雜湊查詢等多種方法從中搜索目標元素。各個方法的工作原理如圖 10-11 所示。</p>
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<p><a class="glightbox" href="../searching_algorithm_revisited.assets/searching_algorithms.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="多種搜尋策略" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></a></p>
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<p align="center"> 圖 10-11 多種搜尋策略 </p>
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<p>上述幾種方法的操作效率與特性如表 10-1 所示。</p>
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<p align="center"> 表 10-1 查詢演算法效率對比 </p>
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<div class="center-table">
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<table>
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<thead>
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<tr>
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<th></th>
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<th>線性搜尋</th>
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<th>二分搜尋</th>
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<th>樹查詢</th>
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<th>雜湊查詢</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>查詢元素</td>
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<td><span class="arithmatex">\(O(n)\)</span></td>
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<td><span class="arithmatex">\(O(\log n)\)</span></td>
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<td><span class="arithmatex">\(O(\log n)\)</span></td>
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<td><span class="arithmatex">\(O(1)\)</span></td>
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</tr>
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<tr>
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<td>插入元素</td>
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<td><span class="arithmatex">\(O(1)\)</span></td>
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<td><span class="arithmatex">\(O(n)\)</span></td>
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<td><span class="arithmatex">\(O(\log n)\)</span></td>
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<td><span class="arithmatex">\(O(1)\)</span></td>
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</tr>
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<tr>
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<td>刪除元素</td>
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<td><span class="arithmatex">\(O(n)\)</span></td>
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<td><span class="arithmatex">\(O(n)\)</span></td>
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<td><span class="arithmatex">\(O(\log n)\)</span></td>
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<td><span class="arithmatex">\(O(1)\)</span></td>
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</tr>
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<tr>
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<td>額外空間</td>
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<td><span class="arithmatex">\(O(1)\)</span></td>
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<td><span class="arithmatex">\(O(1)\)</span></td>
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<td><span class="arithmatex">\(O(n)\)</span></td>
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<td><span class="arithmatex">\(O(n)\)</span></td>
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</tr>
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<tr>
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<td>資料預處理</td>
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<td>/</td>
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<td>排序 <span class="arithmatex">\(O(n \log n)\)</span></td>
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<td>建樹 <span class="arithmatex">\(O(n \log n)\)</span></td>
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<td>建雜湊表 <span class="arithmatex">\(O(n)\)</span></td>
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</tr>
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<tr>
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<td>資料是否有序</td>
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<td>無序</td>
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<td>有序</td>
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<td>有序</td>
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<td>無序</td>
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</tr>
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</tbody>
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</table>
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</div>
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<p>搜尋演算法的選擇還取決於資料體量、搜尋效能要求、資料查詢與更新頻率等。</p>
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<p><strong>線性搜尋</strong></p>
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<ul>
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<li>通用性較好，無須任何資料預處理操作。假如我們僅需查詢一次資料，那麼其他三種方法的資料預處理的時間比線性搜尋的時間還要更長。</li>
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<li>適用於體量較小的資料，此情況下時間複雜度對效率影響較小。</li>
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<li>適用於資料更新頻率較高的場景，因為該方法不需要對資料進行任何額外維護。</li>
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</ul>
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<p><strong>二分搜尋</strong></p>
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<ul>
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<li>適用於大資料量的情況，效率表現穩定，最差時間複雜度為 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
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<li>資料量不能過大，因為儲存陣列需要連續的記憶體空間。</li>
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<li>不適用於高頻增刪資料的場景，因為維護有序陣列的開銷較大。</li>
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</ul>
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<p><strong>雜湊查詢</strong></p>
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<ul>
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<li>適合對查詢效能要求很高的場景，平均時間複雜度為 <span class="arithmatex">\(O(1)\)</span> 。</li>
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<li>不適合需要有序資料或範圍查詢的場景，因為雜湊表無法維護資料的有序性。</li>
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<li>對雜湊函式和雜湊衝突處理策略的依賴性較高，具有較大的效能劣化風險。</li>
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<li>不適合資料量過大的情況，因為雜湊表需要額外空間來最大程度地減少衝突，從而提供良好的查詢效能。</li>
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</ul>
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<p><strong>樹查詢</strong></p>
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<ul>
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<li>適用於海量資料，因為樹節點在記憶體中是分散儲存的。</li>
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<li>適合需要維護有序資料或範圍查詢的場景。</li>
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<li>在持續增刪節點的過程中，二元搜尋樹可能產生傾斜，時間複雜度劣化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
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<li>若使用 AVL 樹或紅黑樹，則各項操作可在 <span class="arithmatex">\(O(\log n)\)</span> 效率下穩定執行，但維護樹平衡的操作會增加額外的開銷。</li>
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