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第 0 章 前書き
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0.1 本書について
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0.2 本書の使い方
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0.3 まとめ
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第 1 章 アルゴリズムを知る
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第 1 章 アルゴリズムを知る
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1.1 アルゴリズムはどこにでもある
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1.2 アルゴリズムとは何か
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第 2 章 計算量解析
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第 2 章 計算量解析
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2.1 アルゴリズムの効率評価
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2.2 反復と再帰
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2.3 時間計算量
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2.4 空間計算量
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2.5 まとめ
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8zm-2 2H5v4h4zM12 2l5.5 9h-11zm0 3.86L10.08 9h3.84zM17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5"/></svg>
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第 3 章 データ構造
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第 3 章 データ構造
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</label>
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<span class="md-ellipsis">
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3.1 データ構造の分類
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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 基本データ型
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
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3.3 数値の符号化 *
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
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3.4 文字の符号化 *
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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3.5 まとめ
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</span>
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</a>
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 4 章 配列と連結リスト
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</span>
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 4 章 配列と連結リスト
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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 配列
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 連結リスト
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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_array_and_linkedlist/list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3 リスト
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</span>
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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 メモリとキャッシュ *
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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 まとめ
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</span>
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</a>
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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38zM6.77 14.32l.37-1.76 8.79 1.85-.37 1.76zm1.16-4.21.76-1.61 8.14 3.78-.76 1.62zm2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37zm4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21zM6.59 18.41v-1.8h8.98v1.8z"/></svg>
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<span class="md-ellipsis">
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第 5 章 スタックとキュー
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 5 章 スタックとキュー
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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 スタック
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</span>
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</a>
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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 キュー
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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 両端キュー
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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 まとめ
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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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<div class="md-nav__link md-nav__container">
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<span class="md-ellipsis">
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第 6 章 ハッシュ表
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 6 章 ハッシュ表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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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 ハッシュ表
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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_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2 ハッシュ衝突
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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_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3 ハッシュアルゴリズム
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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_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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6.4 まとめ
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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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<span class="md-ellipsis">
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第 7 章 木
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</a>
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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 木
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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7.1 二分木
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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 二分木の走査
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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 二分木の配列表現
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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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7.4 二分探索木
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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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7.5 AVL木 *
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7.6 まとめ
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</span>
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</a>
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<a href="../../chapter_heap/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1"/></svg>
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<span class="md-ellipsis">
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第 8 章 ヒープ
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</span>
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</a>
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<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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第 8 章 ヒープ
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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8.1 ヒープ
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</span>
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</a>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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8.2 ヒープ構築操作
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</span>
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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 まとめ
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</span>
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</a>
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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-5.56-9.59zM6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43s.83.16 1.12.43l4.28-2.53zM12 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 1.68-1.68s1.68.75 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 5.25zm2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24z"/></svg>
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<span class="md-ellipsis">
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第 9 章 グラフ
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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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<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 グラフ
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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 グラフ
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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 グラフの基本操作
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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 グラフの走査
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</span>
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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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9.4 まとめ
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</span>
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</a>
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</nav>
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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 4v2H3V4zM3 16v-2h6v2zm0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1z"/></svg>
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<span class="md-ellipsis">
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第 10 章 探索
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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 探索
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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 二分探索
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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/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 二分探索の挿入点
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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 二分探索の境界
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 ハッシュ最適化戦略
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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_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 探索アルゴリズムの再認識
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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 まとめ
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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 6h7v2H2z"/></svg>
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<span class="md-ellipsis">
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第 11 章 ソート
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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 ソート
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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 ソートアルゴリズム
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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 選択ソート
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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 バブルソート
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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 挿入ソート
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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 クイックソート
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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 マージソート
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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 ヒープソート
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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 バケットソート
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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 計数ソート
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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 基数ソート
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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 まとめ
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7zM2 9v6h5V9zm10 0v2H9v2h3v2l3-3zm5 2v2h5v-2zm0 4v2h5v-2z"/></svg>
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<span class="md-ellipsis">
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第 12 章 分割統治
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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 分割統治
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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 分割統治アルゴリズム
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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 分割統治探索戦略
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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 木の構築問題
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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/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.4 ハノイの塔問題
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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 まとめ
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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_15" >
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第 13 章 バックトラッキング
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第 13 章 バックトラッキング
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13.1 バックトラッキングアルゴリズム
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13.2 全順列問題
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13.3 部分集合和問題
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13.4 Nクイーン問題
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13.5 まとめ
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第 14 章 動的計画法
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第 14 章 動的計画法
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<span class="md-ellipsis">
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14.1 動的計画法の初歩
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14.2 DP 問題の特性
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<a href="../dp_solution_pipeline/" class="md-nav__link">
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<span class="md-ellipsis">
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14.3 DP の解法の考え方
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14.4 0-1ナップサック問題
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<a href="../unbounded_knapsack_problem/" class="md-nav__link">
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14.5 完全ナップサック問題
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14.6 編集距離問題
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</span>
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<span class="md-nav__icon md-icon"></span>
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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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14.6 編集距離問題
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目次">
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目次
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1. 動的プログラミングアプローチ
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</span>
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2. コード実装
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3. 空間最適化
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<a href="../summary/" class="md-nav__link">
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<span class="md-ellipsis">
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14.7 まとめ
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</span>
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</a>
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第 15 章 貪欲法
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</span>
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</a>
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第 15 章 貪欲法
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</label>
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15.1 貪欲アルゴリズム
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15.2 分数ナップサック問題
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15.3 最大容量問題
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<a href="../../chapter_greedy/max_product_cutting_problem/" class="md-nav__link">
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15.4 最大積分割問題
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<span class="md-ellipsis">
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15.5 まとめ
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</span>
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</a>
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title="編集"
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 7.1.0 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 2025 Fonticons, Inc.--><path fill="currentColor" 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 0M88 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-10.7-24-24-24z"/></svg>
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<h1 id="146">14.6 編集距離問題<a class="headerlink" href="#146" title="Permanent link">¶</a></h1>
|
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<p>編集距離は、レーベンシュタイン距離とも呼ばれ、一つの文字列を別の文字列に変換するために必要な最小修正回数を指し、情報検索や自然言語処理で2つのシーケンス間の類似度を測定するためによく使用されます。</p>
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<div class="admonition question">
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<p class="admonition-title">Question</p>
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<p>2つの文字列 <span class="arithmatex">\(s\)</span> と <span class="arithmatex">\(t\)</span> が与えられたとき、<span class="arithmatex">\(s\)</span> を <span class="arithmatex">\(t\)</span> に変換するために必要な最小編集回数を返してください。</p>
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<p>文字列に対して3種類の編集を実行できます：文字の挿入、文字の削除、または文字を他の任意の文字に置換。</p>
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</div>
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<p>下の図に示すように、<code>kitten</code> を <code>sitting</code> に変換するには3回の編集が必要で、2回の置換と1回の挿入を含みます。<code>hello</code> を <code>algo</code> に変換するには3ステップが必要で、2回の置換と1回の削除を含みます。</p>
|
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<p><img alt="編集距離の例データ" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_example.png" /></p>
|
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<p align="center"> 図 14-27 編集距離の例データ </p>
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<p><strong>編集距離問題は決定木モデルで自然に説明できます</strong>。文字列は木のノードに対応し、1ラウンドの決定（編集操作）は木のエッジに対応します。</p>
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<p>下の図に示すように、操作に制限がない場合、各ノードは多くのエッジを導出でき、それぞれが1つの操作に対応するため、<code>hello</code> を <code>algo</code> に変換する可能な経路は多数あります。</p>
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<p>決定木の観点から、この問題の目標は、ノード <code>hello</code> とノード <code>algo</code> の間の最短経路を見つけることです。</p>
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<p><img alt="決定木モデルに基づいて表現された編集距離問題" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_decision_tree.png" /></p>
|
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<p align="center"> 図 14-28 決定木モデルに基づいて表現された編集距離問題 </p>
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<h3 id="1">1. 動的プログラミングアプローチ<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
|
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<p><strong>ステップ1：各ラウンドの決定を考え、状態を定義し、それにより <span class="arithmatex">\(dp\)</span> テーブルを得る</strong></p>
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<p>各ラウンドの決定は、文字列 <span class="arithmatex">\(s\)</span> に対して1つの編集操作を実行することを含みます。</p>
|
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<p>編集プロセス中に問題のサイズを段階的に縮小することを目指し、これにより部分問題を構築できます。文字列 <span class="arithmatex">\(s\)</span> と <span class="arithmatex">\(t\)</span> の長さをそれぞれ <span class="arithmatex">\(n\)</span> と <span class="arithmatex">\(m\)</span> とします。まず、両方の文字列の末尾文字 <span class="arithmatex">\(s[n-1]\)</span> と <span class="arithmatex">\(t[m-1]\)</span> を考慮します。</p>
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<ul>
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<li><span class="arithmatex">\(s[n-1]\)</span> と <span class="arithmatex">\(t[m-1]\)</span> が同じ場合、それらをスキップして直接 <span class="arithmatex">\(s[n-2]\)</span> と <span class="arithmatex">\(t[m-2]\)</span> を考慮できます。</li>
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<li><span class="arithmatex">\(s[n-1]\)</span> と <span class="arithmatex">\(t[m-1]\)</span> が異なる場合、<span class="arithmatex">\(s\)</span> に対して1つの編集（挿入、削除、置換）を実行して、2つの文字列の末尾文字を一致させ、それらをスキップしてより小規模な問題を考慮できるようにする必要があります。</li>
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</ul>
|
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<p>したがって、文字列 <span class="arithmatex">\(s\)</span> での各ラウンドの決定（編集操作）は、<span class="arithmatex">\(s\)</span> と <span class="arithmatex">\(t\)</span> でマッチされる残りの文字を変更します。したがって、状態は <span class="arithmatex">\(s\)</span> と <span class="arithmatex">\(t\)</span> で現在考慮されている <span class="arithmatex">\(i\)</span> 番目と <span class="arithmatex">\(j\)</span> 番目の文字であり、<span class="arithmatex">\([i, j]\)</span> と表記されます。</p>
|
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<p>状態 <span class="arithmatex">\([i, j]\)</span> は部分問題に対応します：<strong><span class="arithmatex">\(s\)</span> の最初の <span class="arithmatex">\(i\)</span> 文字を <span class="arithmatex">\(t\)</span> の最初の <span class="arithmatex">\(j\)</span> 文字に変更するために必要な最小編集回数</strong>。</p>
|
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<p>これから、サイズ <span class="arithmatex">\((i+1) \times (j+1)\)</span> の二次元 <span class="arithmatex">\(dp\)</span> テーブルを得ます。</p>
|
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<p><strong>ステップ2：最適部分構造を特定し、状態遷移方程式を導出する</strong></p>
|
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<p>部分問題 <span class="arithmatex">\(dp[i, j]\)</span> を考慮すると、これに対応する2つの文字列の末尾文字は <span class="arithmatex">\(s[i-1]\)</span> と <span class="arithmatex">\(t[j-1]\)</span> であり、下の図に示すように3つのシナリオに分けることができます。</p>
|
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<ol>
|
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<li><span class="arithmatex">\(s[i-1]\)</span> の後に <span class="arithmatex">\(t[j-1]\)</span> を追加すると、残りの部分問題は <span class="arithmatex">\(dp[i, j-1]\)</span> です。</li>
|
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<li><span class="arithmatex">\(s[i-1]\)</span> を削除すると、残りの部分問題は <span class="arithmatex">\(dp[i-1, j]\)</span> です。</li>
|
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<li><span class="arithmatex">\(s[i-1]\)</span> を <span class="arithmatex">\(t[j-1]\)</span> に置換すると、残りの部分問題は <span class="arithmatex">\(dp[i-1, j-1]\)</span> です。</li>
|
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</ol>
|
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<p><img alt="編集距離の状態遷移" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_state_transfer.png" /></p>
|
|
<p align="center"> 図 14-29 編集距離の状態遷移 </p>
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<p>上記の分析に基づいて、最適部分構造を決定できます：<span class="arithmatex">\(dp[i, j]\)</span> の最小編集回数は、<span class="arithmatex">\(dp[i, j-1]\)</span>、<span class="arithmatex">\(dp[i-1, j]\)</span>、<span class="arithmatex">\(dp[i-1, j-1]\)</span> の中の最小値に編集ステップ <span class="arithmatex">\(1\)</span> を加えたものです。対応する状態遷移方程式は：</p>
|
|
<div class="arithmatex">\[
|
|
dp[i, j] = \min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1
|
|
\]</div>
|
|
<p>注意してください。<strong><span class="arithmatex">\(s[i-1]\)</span> と <span class="arithmatex">\(t[j-1]\)</span> が同じ場合、現在の文字に対して編集は必要ありません</strong>。この場合、状態遷移方程式は：</p>
|
|
<div class="arithmatex">\[
|
|
dp[i, j] = dp[i-1, j-1]
|
|
\]</div>
|
|
<p><strong>ステップ3：境界条件と状態遷移の順序を決定する</strong></p>
|
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<p>両方の文字列が空の場合、編集回数は <span class="arithmatex">\(0\)</span> です。つまり、<span class="arithmatex">\(dp[0, 0] = 0\)</span> です。<span class="arithmatex">\(s\)</span> が空で <span class="arithmatex">\(t\)</span> が空でない場合、最小編集回数は <span class="arithmatex">\(t\)</span> の長さに等しく、つまり最初の行 <span class="arithmatex">\(dp[0, j] = j\)</span> です。<span class="arithmatex">\(s\)</span> が空でなく <span class="arithmatex">\(t\)</span> が空の場合、最小編集回数は <span class="arithmatex">\(s\)</span> の長さに等しく、つまり最初の列 <span class="arithmatex">\(dp[i, 0] = i\)</span> です。</p>
|
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<p>状態遷移方程式を観察すると、<span class="arithmatex">\(dp[i, j]\)</span> の解決は左、上、左上の解に依存するため、二重ループを使用して正しい順序で <span class="arithmatex">\(dp\)</span> テーブル全体を走査できます。</p>
|
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<h3 id="2">2. コード実装<a class="headerlink" href="#2" title="Permanent link">¶</a></h3>
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<div class="highlight"><span class="filename">edit_distance.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">edit_distance_dp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">"""編集距離：動的プログラミング"""</span>
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
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<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
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<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="c1"># 状態遷移：最初の行と最初の列</span>
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<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
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<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
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<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="c1"># 状態遷移：残りの行と列</span>
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<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
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<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="c1"># 2 つの文字が等しい場合、これら 2 つの文字をスキップ</span>
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<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
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<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="c1"># 最小編集数 = 3 つの操作（挿入、削除、置換）からの最小編集数 + 1</span>
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<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span>
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<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a> <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 編集距離：動的プログラミング */</span>
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">();</span>
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<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>></span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
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<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="c1">// 状態遷移：最初の行と最初の列</span>
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<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
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<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
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<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="c1">// 状態遷移：残りの行と列</span>
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<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="c1">// 2つの文字が等しい場合、これら2つの文字をスキップ</span>
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<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
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<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="c1">// 最小編集数 = 3つの操作（挿入、削除、置換）からの最小編集数 + 1</span>
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<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
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<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">];</span>
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<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 編集距離：動的プログラミング */</span>
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span><span class="p">();</span>
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|
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
|
|
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w"> </span><span class="c1">// 状態遷移：最初の行と最初の列</span>
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|
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
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<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
|
|
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
|
|
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w"> </span><span class="c1">// 状態遷移：残りの行と列</span>
|
|
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w"> </span><span class="c1">// 2つの文字が等しい場合、これら2つの文字をスキップ</span>
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<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
|
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w"> </span><span class="c1">// 最小編集数 = 3つの操作（挿入、削除、置換）からの最小編集数 + 1</span>
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<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
|
|
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w"> </span><span class="p">}</span>
|
|
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">][</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
|
|
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">edit_distance</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">EditDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
|
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</div>
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<div class="tabbed-block">
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|
<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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|
<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">edit_distance_dp</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
|
|
<div class="highlight"><span class="filename">edit_distance.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">editDistanceDP</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">edit_distance_dp</span><span class="p">}</span>
|
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</code></pre></div>
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</div>
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</div>
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</div>
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<p>下の図に示すように、編集距離問題の状態遷移プロセスはナップサック問題と非常に似ており、二次元グリッドを埋めることと見なすことができます。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:15"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><input id="__tabbed_2_15" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1"><1></label><label for="__tabbed_2_2"><2></label><label for="__tabbed_2_3"><3></label><label for="__tabbed_2_4"><4></label><label for="__tabbed_2_5"><5></label><label for="__tabbed_2_6"><6></label><label for="__tabbed_2_7"><7></label><label for="__tabbed_2_8"><8></label><label for="__tabbed_2_9"><9></label><label for="__tabbed_2_10"><10></label><label for="__tabbed_2_11"><11></label><label for="__tabbed_2_12"><12></label><label for="__tabbed_2_13"><13></label><label for="__tabbed_2_14"><14></label><label for="__tabbed_2_15"><15></label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<p><img alt="編集距離の動的プログラミングプロセス" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step1.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step2" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step2.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step3" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step3.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step4" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step4.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step5" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step5.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step6" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step6.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step7" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step7.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step8" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step8.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step9" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step9.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step10" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step10.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step11" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step11.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step12" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step12.png" /></p>
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</div>
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<div class="tabbed-block">
|
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<p><img alt="edit_distance_dp_step13" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step13.png" /></p>
|
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</div>
|
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<div class="tabbed-block">
|
|
<p><img alt="edit_distance_dp_step14" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step14.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="edit_distance_dp_step15" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step15.png" /></p>
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</div>
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</div>
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</div>
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<p align="center"> 図 14-30 編集距離の動的プログラミングプロセス </p>
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<h3 id="3">3. 空間最適化<a class="headerlink" href="#3" title="Permanent link">¶</a></h3>
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<p><span class="arithmatex">\(dp[i, j]\)</span> は上の <span class="arithmatex">\(dp[i-1, j]\)</span>、左の <span class="arithmatex">\(dp[i, j-1]\)</span>、左上の <span class="arithmatex">\(dp[i-1, j-1]\)</span> の解から導出され、直接走査では左上の解 <span class="arithmatex">\(dp[i-1, j-1]\)</span> が失われ、逆走査では事前に <span class="arithmatex">\(dp[i, j-1]\)</span> を構築できないため、どちらの走査順序も実行可能ではありません。</p>
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<p>この理由で、変数 <code>leftup</code> を使用して左上の <span class="arithmatex">\(dp[i-1, j-1]\)</span> からの解を一時的に保存し、左と上の解のみを考慮すればよくなります。この状況は無制限ナップサック問題と似ており、直接走査が可能です。コードは以下の通りです：</p>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">edit_distance_dp_comp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
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<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w"> </span><span class="sd">"""編集距離：空間最適化動的プログラミング"""</span>
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<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
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<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a> <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a> <span class="c1"># 状態遷移：最初の行</span>
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<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
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<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a> <span class="c1"># 状態遷移：残りの行</span>
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<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a> <span class="c1"># 状態遷移：最初の列</span>
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<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a> <span class="n">leftup</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c1"># dp[i-1, j-1] を一時的に保存</span>
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<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
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<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a> <span class="c1"># 状態遷移：残りの列</span>
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<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a> <span class="n">temp</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
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<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a> <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
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<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a> <span class="c1"># 2 つの文字が等しい場合、これら 2 つの文字をスキップ</span>
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<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">leftup</span>
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<a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a> <span class="c1"># 最小編集数 = 3 つの操作（挿入、削除、置換）からの最小編集数 + 1</span>
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<a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">leftup</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
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<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a> <span class="n">leftup</span> <span class="o">=</span> <span class="n">temp</span> <span class="c1"># 次の dp[i-1, j-1] のために更新</span>
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<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a> <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">editDistanceDPComp</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 編集距離：空間最適化動的プログラミング */</span>
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<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span><span class="p">();</span>
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<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w"> </span><span class="c1">// 状態遷移：最初の行</span>
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<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
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<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w"> </span><span class="c1">// 状態遷移：残りの行</span>
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<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w"> </span><span class="c1">// 状態遷移：最初の列</span>
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<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// dp[i-1, j-1] を一時的に格納</span>
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<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
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<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w"> </span><span class="c1">// 状態遷移：残りの列</span>
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<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w"> </span><span class="c1">// 2つの文字が等しい場合、これら2つの文字をスキップ</span>
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<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
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<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w"> </span><span class="c1">// 最小編集数 = 3つの操作（挿入、削除、置換）からの最小編集数 + 1</span>
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<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
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<a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 次のラウンドの dp[i-1, j-1] のために更新</span>
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<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
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<a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">edit_distance</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">EditDistanceDPComp</span><span class="p">}</span>
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<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">editDistanceDPComp</span><span class="p">}</span>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">editDistanceDPComp</span><span class="p">}</span>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">editDistanceDPComp</span><span class="p">}</span>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">editDistanceDPComp</span><span class="p">}</span>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">editDistanceDPComp</span><span class="p">}</span>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">edit_distance_dp_comp</span><span class="p">}</span>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">editDistanceDPComp</span><span class="p">}</span>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">editDistanceDPComp</span><span class="p">}</span>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">edit_distance.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">edit_distance_dp_comp</span><span class="p">}</span>
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