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<span class="md-ellipsis">
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はじめに
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はじめに
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第 0 章 前書き
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第 0 章 前書き
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0.1 この本について
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</span>
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</a>
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0.2 本書の使い方
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</span>
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</a>
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0.3 まとめ
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第 1 章 アルゴリズムとの出会い
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第 1 章 アルゴリズムとの出会い
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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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1.3 まとめ
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<span class="md-ellipsis">
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第 2 章 計算量解析
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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第 2 章 計算量解析
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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2.1 アルゴリズム効率評価
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<li class="md-nav__item">
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2.2 反復と再帰
|
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|
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|
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|
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</span>
|
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|
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|
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|
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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_computational_complexity/time_complexity/" class="md-nav__link">
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|
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|
||
2.3 時間計算量
|
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|
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|
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|
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</span>
|
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|
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|
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|
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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_computational_complexity/space_complexity/" class="md-nav__link">
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|
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|
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2.4 空間計算量
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<li class="md-nav__item">
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|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
2.5 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
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|
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|
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||
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||
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||
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|
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|
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|
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|
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|
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|
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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_5" >
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|
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_data_structure/" class="md-nav__link ">
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
第 3 章 データ構造
|
||
|
||
|
||
|
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</span>
|
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|
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|
||
</a>
|
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<label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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<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_5_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_5">
|
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<span class="md-nav__icon md-icon"></span>
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第 3 章 データ構造
|
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|
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|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
3.1 データ構造の分類
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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||
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
3.2 基本データ型
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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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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|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
3.3 数値エンコーディング *
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
||
<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
3.4 文字エンコーディング *
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
|
||
|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
3.5 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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|
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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_6" >
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|
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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">
|
||
|
||
|
||
第 4 章 配列と連結リスト
|
||
|
||
|
||
|
||
</span>
|
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|
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|
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|
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</a>
|
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<label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
|
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<span class="md-nav__icon md-icon"></span>
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</div>
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_6">
|
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<span class="md-nav__icon md-icon"></span>
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|
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第 4 章 配列と連結リスト
|
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|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
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|
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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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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
4.1 配列
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
4.2 連結リスト
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
4.3 リスト
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
4.4 メモリとキャッシュ *
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
4.5 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
|
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|
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|
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<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
|
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|
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|
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|
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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>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
第 5 章 スタックとキュー
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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|
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|
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<label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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|
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</div>
|
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|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_7">
|
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<span class="md-nav__icon md-icon"></span>
|
||
|
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|
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第 5 章 スタックとキュー
|
||
|
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|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
5.1 スタック
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
||
|
||
|
||
|
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|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
5.2 キュー
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
5.3 両端キュー
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
5.4 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item md-nav__item--nested">
|
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|
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|
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|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
|
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|
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|
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<div class="md-nav__link md-nav__container">
|
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<a href="../../chapter_hashing/" class="md-nav__link ">
|
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|
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|
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|
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21zm-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54zM19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
第 6 章 ハッシュ表
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
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|
||
<label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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|
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</div>
|
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_8">
|
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<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
第 6 章 ハッシュ表
|
||
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
6.1 ハッシュ表
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
6.2 ハッシュ衝突
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
6.3 ハッシュアルゴリズム
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
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|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
||
6.4 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</a>
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|
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</ul>
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|
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第 7 章 木
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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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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
7.1 二分木
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
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|
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|
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|
||
7.2 二分木の走査
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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|
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|
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|
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|
||
|
||
|
||
7.3 木の配列表現
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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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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|
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|
||
|
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<span class="md-ellipsis">
|
||
|
||
|
||
7.4 二分探索木
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
||
|
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|
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|
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|
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|
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|
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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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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
7.5 AVL木 *
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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||
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/summary/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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|
||
|
||
7.6 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
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|
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|
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|
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|
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</ul>
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</nav>
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|
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||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_heap/" class="md-nav__link ">
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|
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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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|
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<span class="md-ellipsis">
|
||
|
||
|
||
第 8 章 ヒープ
|
||
|
||
|
||
|
||
</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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|
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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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|
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第 8 章 ヒープ
|
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|
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|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_heap/heap/" class="md-nav__link">
|
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|
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
8.1 ヒープ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||
<li class="md-nav__item">
|
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
8.2 ヒープの構築
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/top_k/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
8.3 Top-k問題
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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|
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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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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
8.4 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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</ul>
|
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</nav>
|
||
|
||
</li>
|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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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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|
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<span class="md-ellipsis">
|
||
|
||
|
||
第 9 章 グラフ
|
||
|
||
|
||
|
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</span>
|
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|
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|
||
|
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</a>
|
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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|
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|
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第 9 章 グラフ
|
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|
||
|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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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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|
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.1 グラフ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
||
|
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|
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|
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|
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|
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|
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|
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|
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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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|
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.2 グラフの基本操作
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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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">
|
||
|
||
|
||
9.3 グラフの走査
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
||
|
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|
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|
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|
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|
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|
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|
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|
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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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|
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|
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|
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<span class="md-ellipsis">
|
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|
||
|
||
9.4 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</a>
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</li>
|
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||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_searching/" class="md-nav__link ">
|
||
|
||
|
||
|
||
<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>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
第 10 章 探索
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_12">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
第 10 章 探索
|
||
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.1 二分探索
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.2 二分探索挿入点
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.3 二分探索の境界
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.4 ハッシュ最適化戦略
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.5 探索アルゴリズム再考
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.6 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_sorting/" class="md-nav__link ">
|
||
|
||
|
||
|
||
<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>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
第 11 章 ソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_13">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
第 11 章 ソート
|
||
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.1 ソートアルゴリズム
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.2 選択ソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.3 バブルソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.4 挿入ソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.5 クイックソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.6 マージソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.7 ヒープソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.8 バケットソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.9 計数ソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.10 基数ソート
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.11 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
|
||
|
||
|
||
|
||
<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>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
第 12 章 分割統治
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
<label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_14">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
第 12 章 分割統治
|
||
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
12.1 分割統治アルゴリズム
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
12.2 分割統治探索戦略
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
12.3 二分木構築問題
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
12.4 ハノイの塔問題
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
12.5 まとめ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
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第 13 章 バックトラッキング
|
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|
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|
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|
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第 13 章 バックトラッキング
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|
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|
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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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|
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|
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|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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13.2 順列問題
|
||
|
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|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
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|
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13.3 部分集合和問題
|
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|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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13.4 Nクイーン問題
|
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<span class="md-nav__icon md-icon"></span>
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13.4 Nクイーン問題
|
||
|
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|
||
|
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</span>
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</a>
|
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目次
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|
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1. 行ごとの配置戦略
|
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|
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2. 列と対角線の剪定
|
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3. コード実装
|
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|
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|
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13.5 まとめ
|
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|
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|
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|
||
</span>
|
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|
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|
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|
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<span class="md-ellipsis">
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第 14 章 動的プログラミング
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第 14 章 動的プログラミング
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|
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14.1 動的プログラミング入門
|
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|
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|
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|
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</span>
|
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|
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|
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</a>
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|
||
14.2 DP問題の特性
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
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|
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|
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|
||
14.3 DP問題解決アプローチ
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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<li class="md-nav__item">
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|
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|
||
14.4 0-1ナップサック問題
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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|
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14.5 無制限ナップサック問題
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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|
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|
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14.6 編集距離問題
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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14.7 まとめ
|
||
|
||
|
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|
||
</span>
|
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|
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|
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|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
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第 15 章 貪欲法
|
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|
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|
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|
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第 15 章 貪欲法
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|
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15.1 貪欲アルゴリズム
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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15.2 分数ナップサック問題
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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15.3 最大容量問題
|
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|
||
|
||
|
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</span>
|
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|
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|
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|
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</a>
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</li>
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15.4 最大積切断問題
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<h1 id="134-n">13.4 Nクイーン問題<a class="headerlink" href="#134-n" title="Permanent link">¶</a></h1>
|
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<div class="admonition question">
|
||
<p class="admonition-title">Question</p>
|
||
<p>チェスのルールによると、クイーンは同じ行、列、または対角線上の駒を攻撃できます。<span class="arithmatex">\(n\)</span> 個のクイーンと <span class="arithmatex">\(n \times n\)</span> のチェスボードが与えられた場合、2つのクイーンが互いに攻撃できない配置を見つけてください。</p>
|
||
</div>
|
||
<p>以下の図に示すように、<span class="arithmatex">\(n = 4\)</span> の場合、2つの解があります。バックトラッキングアルゴリズムの観点から、<span class="arithmatex">\(n \times n\)</span> のチェスボードには <span class="arithmatex">\(n^2\)</span> 個のマスがあり、すべての可能な選択肢 <code>choices</code> を示しています。チェスボードの状態 <code>state</code> は、各クイーンが配置されるにつれて継続的に変化します。</p>
|
||
<p><a class="glightbox" href="../n_queens_problem.assets/solution_4_queens.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="4クイーン問題の解" class="animation-figure" src="../n_queens_problem.assets/solution_4_queens.png" /></a></p>
|
||
<p align="center"> 図 13-15 4クイーン問題の解 </p>
|
||
|
||
<p>以下の図は、この問題の3つの制約を示しています：<strong>複数のクイーンは同じ行、列、または対角線を占有できません</strong>。対角線は主対角線 <code>\</code> と副対角線 <code>/</code> に分かれることに注意することが重要です。</p>
|
||
<p><a class="glightbox" href="../n_queens_problem.assets/n_queens_constraints.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Nクイーン問題の制約" class="animation-figure" src="../n_queens_problem.assets/n_queens_constraints.png" /></a></p>
|
||
<p align="center"> 図 13-16 Nクイーン問題の制約 </p>
|
||
|
||
<h3 id="1">1. 行ごとの配置戦略<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
|
||
<p>クイーンの数がチェスボードの行数と等しく、どちらも <span class="arithmatex">\(n\)</span> であるため、**チェスボードの各行には1つのクイーンのみが配置できることが**容易に結論付けられます。</p>
|
||
<p>これは、行ごとの配置戦略を採用できることを意味します：最初の行から開始して、最後の行に到達するまで行ごとに1つのクイーンを配置します。</p>
|
||
<p>以下の図は、4クイーン問題の行ごとの配置プロセスを示しています。スペースの制限により、図は最初の行の1つの検索分岐のみを展開し、列と対角線の制約を満たさない配置を剪定します。</p>
|
||
<p><a class="glightbox" href="../n_queens_problem.assets/n_queens_placing.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="行ごとの配置戦略" class="animation-figure" src="../n_queens_problem.assets/n_queens_placing.png" /></a></p>
|
||
<p align="center"> 図 13-17 行ごとの配置戦略 </p>
|
||
|
||
<p>本質的に、<strong>行ごとの配置戦略は剪定関数として機能し</strong>、同じ行に複数のクイーンを配置するすべての検索分岐を除去します。</p>
|
||
<h3 id="2">2. 列と対角線の剪定<a class="headerlink" href="#2" title="Permanent link">¶</a></h3>
|
||
<p>列の制約を満たすために、長さ <span class="arithmatex">\(n\)</span> のブール配列 <code>cols</code> を使用して、各列にクイーンが占有されているかどうかを追跡できます。各配置決定の前に、<code>cols</code> を使用してすでにクイーンがある列を剪定し、バックトラッキング中に動的に更新されます。</p>
|
||
<div class="admonition tip">
|
||
<p class="admonition-title">Tip</p>
|
||
<p>行列の原点は左上隅にあり、行インデックスは上から下に増加し、列インデックスは左から右に増加することに注意してください。</p>
|
||
</div>
|
||
<p>対角線の制約はどうでしょうか？チェスボード上の特定のセルの行と列のインデックスを <span class="arithmatex">\((row, col)\)</span> とします。特定の主対角線を選択することで、その対角線上のすべてのセルで差 <span class="arithmatex">\(row - col\)</span> が同じであることに気付きます。<strong>つまり、<span class="arithmatex">\(row - col\)</span> は主対角線上で定数値です</strong>。</p>
|
||
<p>言い換えると、2つのセルが <span class="arithmatex">\(row_1 - col_1 = row_2 - col_2\)</span> を満たす場合、それらは確実に同じ主対角線上にあります。このパターンを使用して、以下の図に示す配列 <code>diags1</code> を利用して、クイーンが主対角線上にあるかどうかを追跡できます。</p>
|
||
<p>同様に、<strong><span class="arithmatex">\(row + col\)</span> の和は副対角線上のすべてのセルで定数値です</strong>。配列 <code>diags2</code> を使用して副対角線の制約も処理できます。</p>
|
||
<p><a class="glightbox" href="../n_queens_problem.assets/n_queens_cols_diagonals.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="列と対角線の制約の処理" class="animation-figure" src="../n_queens_problem.assets/n_queens_cols_diagonals.png" /></a></p>
|
||
<p align="center"> 図 13-18 列と対角線の制約の処理 </p>
|
||
|
||
<h3 id="3">3. コード実装<a class="headerlink" href="#3" title="Permanent link">¶</a></h3>
|
||
<p><span class="arithmatex">\(n\)</span> 次元の正方行列では、<span class="arithmatex">\(row - col\)</span> の範囲は <span class="arithmatex">\([-n + 1, n - 1]\)</span> で、<span class="arithmatex">\(row + col\)</span> の範囲は <span class="arithmatex">\([0, 2n - 2]\)</span> であることに注意してください。したがって、主対角線と副対角線の数はどちらも <span class="arithmatex">\(2n - 1\)</span> で、配列 <code>diags1</code> と <code>diags2</code> の長さは <span class="arithmatex">\(2n - 1\)</span> です。</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="1:14"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label><label for="__tabbed_1_14">Zig</label></div>
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<div class="tabbed-content">
|
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">(</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="n">row</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
|
||
<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="nb">int</span><span class="p">,</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]],</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]],</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="n">cols</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="n">diags1</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="n">diags2</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="p">):</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="sd">"""バックトラッキングアルゴリズム：n クイーン"""</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="c1"># すべての行が配置されたら、解を記録</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="k">if</span> <span class="n">row</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="n">row</span><span class="p">)</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">state</span><span class="p">])</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="k">return</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="c1"># すべての列を走査</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="c1"># セルに対応する主対角線と副対角線を計算</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="n">diag1</span> <span class="o">=</span> <span class="n">row</span> <span class="o">-</span> <span class="n">col</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a> <span class="n">diag2</span> <span class="o">=</span> <span class="n">row</span> <span class="o">+</span> <span class="n">col</span>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a> <span class="c1"># 枝刈り：セルの列、主対角線、副対角線にクイーンを配置しない</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]:</span>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a> <span class="c1"># 試行：セルにクイーンを配置</span>
|
||
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"Q"</span>
|
||
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
|
||
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a> <span class="c1"># 次の行を配置</span>
|
||
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a> <span class="n">backtrack</span><span class="p">(</span><span class="n">row</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">)</span>
|
||
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a> <span class="c1"># 撤回：セルを空のスポットに復元</span>
|
||
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a> <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"#"</span>
|
||
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
|
||
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>
|
||
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="k">def</span><span class="w"> </span><span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]]:</span>
|
||
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="w"> </span><span class="sd">"""n クイーンを解く"""</span>
|
||
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a> <span class="c1"># n*n サイズのチェスボードを初期化、'Q' はクイーンを表し、'#' は空のスポットを表す</span>
|
||
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[[</span><span class="s2">"#"</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="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="p">)]</span>
|
||
<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a> <span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span> <span class="c1"># クイーンがある列を記録</span>
|
||
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a> <span class="n">diags1</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># クイーンがある主対角線を記録</span>
|
||
<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a> <span class="n">diags2</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># クイーンがある副対角線を記録</span>
|
||
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a> <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a> <span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">)</span>
|
||
<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>
|
||
<a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a> <span class="k">return</span> <span class="n">res</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* バックトラッキングアルゴリズム：n クイーン */</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>></span><span class="w"> </span><span class="o">&</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>>></span><span class="w"> </span><span class="o">&</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">cols</span><span class="p">,</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="o">&</span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="c1">// すべての行が配置されたら、解を記録</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</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="p">{</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="c1">// すべての列を走査</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></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">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</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">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="c1">// セルに対応する主対角線と副対角線を計算</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </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>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="c1">// 剪定：セルの列、主対角線、副対角線にクイーンを配置することを許可しない</span>
|
||
<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="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="c1">// 試行：セルにクイーンを配置</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span><span class="p">;</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w"> </span><span class="c1">// 次の行を配置</span>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</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">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w"> </span><span class="c1">// 回退：セルを空のスポットに復元</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span><span class="p">;</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
|
||
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="p">}</span>
|
||
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
|
||
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* n クイーンを解く */</span>
|
||
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>>></span><span class="w"> </span><span class="n">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w"> </span><span class="c1">// n*n サイズのチェスボードを初期化、'Q' はクイーンを表し、'#' は空のスポットを表す</span>
|
||
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></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="n">string</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">></span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="s">"#"</span><span class="p">));</span>
|
||
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">cols</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// クイーンのある列を記録</span>
|
||
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">diags1</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </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="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// クイーンのある主対角線を記録</span>
|
||
<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w"> </span><span class="n">vector</span><span class="o"><</span><span class="kt">bool</span><span class="o">></span><span class="w"> </span><span class="n">diags2</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </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="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// クイーンのある副対角線を記録</span>
|
||
<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></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="n">vector</span><span class="o"><</span><span class="n">string</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a>
|
||
<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
|
||
<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a>
|
||
<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* バックトラッキングアルゴリズム：n クイーン */</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="c1">// すべての行が配置されたら、解を記録</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</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="p">{</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>></span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">></span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </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="n">copyState</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">(</span><span class="n">sRow</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="p">}</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">copyState</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="k">return</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="p">}</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w"> </span><span class="c1">// すべての列を走査</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">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</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">col</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="c1">// セルに対応する主対角線と副対角線を計算</span>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </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>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w"> </span><span class="c1">// 剪定：セルの列、主対角線、副対角線にクイーンを配置することを許可しない</span>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w"> </span><span class="c1">// 試行：セルにクイーンを配置</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">"Q"</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="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</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="c1">// 次の行を配置</span>
|
||
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</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">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w"> </span><span class="c1">// 回退：セルを空のスポットに復元</span>
|
||
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">"#"</span><span class="p">);</span>
|
||
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w"> </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
|
||
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="p">}</span>
|
||
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a>
|
||
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="cm">/* n クイーンを解く */</span>
|
||
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>>></span><span class="w"> </span><span class="nf">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||
<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w"> </span><span class="c1">// n*n サイズのチェスボードを初期化、'Q' はクイーンを表し、'#' は空のスポットを表す</span>
|
||
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>></span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
||
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">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-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">></span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
||
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></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">0</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">n</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-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="w"> </span><span class="n">row</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="s">"#"</span><span class="p">);</span>
|
||
<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
|
||
<a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="w"> </span><span class="p">}</span>
|
||
<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</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">boolean</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// クイーンのある列を記録</span>
|
||
<a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</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">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </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="p">;</span><span class="w"> </span><span class="c1">// クイーンのある主対角線を記録</span>
|
||
<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</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">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </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="p">;</span><span class="w"> </span><span class="c1">// クイーンのある副対角線を記録</span>
|
||
<a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="w"> </span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">List</span><span class="o"><</span><span class="n">String</span><span class="o">>>></span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o"><></span><span class="p">();</span>
|
||
<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a>
|
||
<a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
|
||
<a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a>
|
||
<a id="__codelineno-2-50" name="__codelineno-2-50" href="#__codelineno-2-50"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
|
||
<a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">n_queens</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">Backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="na">[class]</span><span class="p">{</span><span class="n">n_queens</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">NQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
|
||
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
|
||
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
|
||
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
|
||
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a>
|
||
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
|
||
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></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">n_queens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a>
|
||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a>
|
||
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a>
|
||
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></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">n_queens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">n_queens.zig</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></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">backtrack</span><span class="p">}</span>
|
||
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>
|
||
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></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">nQueens</span><span class="p">}</span>
|
||
</code></pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p><span class="arithmatex">\(n\)</span> 個のクイーンを行ごとに配置し、列の制約を考慮して、最初の行から最後の行まで、<span class="arithmatex">\(n\)</span>、<span class="arithmatex">\(n-1\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(1\)</span> の選択肢があり、<span class="arithmatex">\(O(n!)\)</span> 時間を使用します。解を記録する際、行列 <code>state</code> をコピーして <code>res</code> に追加する必要があり、コピー操作は <span class="arithmatex">\(O(n^2)\)</span> 時間を使用します。したがって、<strong>全体の時間計算量は <span class="arithmatex">\(O(n! \cdot n^2)\)</span> です</strong>。実際には、対角線制約に基づく剪定により検索空間を大幅に削減できるため、多くの場合、検索効率は上記の時間計算量よりも優れています。</p>
|
||
<p>配列 <code>state</code> は <span class="arithmatex">\(O(n^2)\)</span> 空間を使用し、配列 <code>cols</code>、<code>diags1</code>、<code>diags2</code> はそれぞれ <span class="arithmatex">\(O(n)\)</span> 空間を使用します。最大再帰深度は <span class="arithmatex">\(n\)</span> で、<span class="arithmatex">\(O(n)\)</span> のスタックフレーム空間を使用します。したがって、<strong>空間計算量は <span class="arithmatex">\(O(n^2)\)</span> です</strong>。</p>
|
||
|
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