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第 8 章 &nbsp; ヒープ
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第 9 章 &nbsp; グラフ
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第 9 章 &nbsp; グラフ
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第 10 章 &nbsp; 探索
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10.2 &nbsp; 二分探索挿入点
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10.3 &nbsp; 二分探索の境界
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10.4 &nbsp; ハッシュ最適化戦略
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10.5 &nbsp; 探索アルゴリズム再考
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第 11 章 &nbsp; ソート
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11.8 &nbsp; バケットソート
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第 12 章 &nbsp; 分割統治
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12.1 &nbsp; 分割統治アルゴリズム
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12.3 &nbsp; 二分木構築問題
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12.4 &nbsp; ハノイの塔問題
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第 13 章 &nbsp; バックトラッキング
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13.1 &nbsp; バックトラッキングアルゴリズム
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13.2 &nbsp; 順列問題
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13.3 &nbsp; 部分集合和問題
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13.4 &nbsp; Nクイーン問題
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第 14 章 &nbsp; 動的プログラミング
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14.1 &nbsp; 動的プログラミング入門
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14.2 &nbsp; DP問題の特性
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14.3 &nbsp; DP問題解決アプローチ
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14.4 &nbsp; 0-1ナップサック問題
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14.5 &nbsp; 無制限ナップサック問題
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14.6 &nbsp; 編集距離問題
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<h1 id="11">1.1 &nbsp; アルゴリズムは至る所にある<a class="headerlink" href="#11" title="Permanent link">&para;</a></h1>
<p>「アルゴリズム」という言葉を聞くと、自然に数学を思い浮かべます。しかし、多くのアルゴリズムは複雑な数学を含まず、基本的な論理により多く依存しており、これは私たちの日常生活の至る所で見ることができます。</p>
<p>アルゴリズムについて正式に議論を始める前に、興味深い事実を共有する価値があります。<strong>あなたは無意識のうちに多くのアルゴリズムを学び、日常生活でそれらを応用することに慣れています</strong>。ここで、この点を証明するためにいくつかの具体的な例を挙げます。</p>
<p><strong>例1：辞書の引き方</strong>。英語の辞書では、単語がアルファベット順に並んでいます。<span class="arithmatex">\(r\)</span>で始まる単語を探していると仮定すると、通常は以下の方法で行います：</p>
<ol>
<li>辞書を大体半分ぐらいのところで開き、そのページの最初の語彙を確認します。例えば<span class="arithmatex">\(m\)</span>で始まる文字だとしましょう。</li>
<li><span class="arithmatex">\(r\)</span>はアルファベットで<span class="arithmatex">\(m\)</span>の後に来るので、前半を無視して、探索空間を後半に絞ります。</li>
<li><span class="arithmatex">\(r\)</span>で始まる単語を見つけるまで、ステップ<code>1.</code>と<code>2.</code>を繰り返します。</li>
</ol>
<div class="tabbed-set tabbed-alternate" data-tabs="1:5"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label></div>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="辞書を引く過程" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="辞書での二分探索ステップ2" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="辞書での二分探索ステップ3" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="辞書での二分探索ステップ4" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" /></a></p>
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<p><a class="glightbox" href="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="辞書での二分探索ステップ5" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" /></a></p>
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<p align="center"> 図 1-1 &nbsp; 辞書を引く過程 </p>
<p>辞書を引くことは、小学生にとって必須のスキルですが、実際には有名な「二分探索」アルゴリズムです。データ構造の観点から、辞書をソートされた「配列」と考えることができます。アルゴリズムの観点から、辞書で単語を探すために取られる一連の行動は、「二分探索」アルゴリズムと見なすことができます。</p>
<p><strong>例2：トランプの整理</strong>。トランプをプレイするとき、手札を昇順に並べる必要があります。以下の過程で示されます。</p>
<ol>
<li>トランプを「整列済み」と「未整列」のセクションに分けます。最初は一番左のカードが既に整列していると仮定します。</li>
<li>未整列セクションからカードを1枚取り出し、整列済みセクションの正しい位置に挿入します。この後、左端の2枚のカードが整列します。</li>
<li>すべてのカードが整列するまで、ステップ<code>2</code>を繰り返します。</li>
</ol>
<p><a class="glightbox" href="../algorithms_are_everywhere.assets/playing_cards_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="トランプの整理過程" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></a></p>
<p align="center"> 図 1-2 &nbsp; トランプの整理過程 </p>
<p>上記のトランプを整理する方法は、実質的に「挿入ソート」アルゴリズムであり、小さなデータセットに対して非常に効率的です。多くのプログラミング言語のソート関数には挿入ソートが含まれています。</p>
<p><strong>例3：お釣りの計算</strong>。スーパーマーケットで<span class="arithmatex">\(69\)</span>の買い物をしたと仮定します。レジ係に<span class="arithmatex">\(100\)</span>を渡すと、<span class="arithmatex">\(31\)</span>のお釣りを提供する必要があります。この過程は以下の図で明確に理解できます。</p>
<ol>
<li>選択肢は<span class="arithmatex">\(31\)</span>以下の価値のある通貨で、<span class="arithmatex">\(1\)</span>、<span class="arithmatex">\(5\)</span>、<span class="arithmatex">\(10\)</span>、<span class="arithmatex">\(20\)</span>が含まれます。</li>
<li>選択肢から最大の<span class="arithmatex">\(20\)</span>を取り出し、<span class="arithmatex">\(31 - 20 = 11\)</span>が残ります。</li>
<li>残りの選択肢から最大の<span class="arithmatex">\(10\)</span>を取り出し、<span class="arithmatex">\(11 - 10 = 1\)</span>が残ります。</li>
<li>残りの選択肢から最大の<span class="arithmatex">\(1\)</span>を取り出し、<span class="arithmatex">\(1 - 1 = 0\)</span>が残ります。</li>
<li>お釣りの計算が完了し、解答は<span class="arithmatex">\(20 + 10 + 1 = 31\)</span>です。</li>
</ol>
<p><a class="glightbox" href="../algorithms_are_everywhere.assets/greedy_change.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="お釣りの計算過程" class="animation-figure" src="../algorithms_are_everywhere.assets/greedy_change.png" /></a></p>
<p align="center"> 図 1-3 &nbsp; お釣りの計算過程 </p>
<p>記述されたステップでは、利用可能な最大の額面を使用して各段階で最良の選択肢を選ぶことで、効果的なお釣り計算戦略につながります。データ構造とアルゴリズムの観点から、このアプローチは「貪欲」アルゴリズムとして知られています。</p>
<p>料理の準備から宇宙旅行まで、ほぼすべての問題解決にはアルゴリズムが関わっています。コンピュータの出現により、メモリにデータ構造を格納し、CPUとGPUを呼び出してアルゴリズムを実行するコードを書くことができるようになりました。このようにして、現実世界の問題をコンピュータに移し、より効率的な方法でさまざまな複雑な問題を解決できます。</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>データ構造、アルゴリズム、配列、二分探索などの概念についてまだ混乱している場合は、読み続けることをお勧めします。この本は、データ構造とアルゴリズムの理解の領域へと優しく導いてくれるでしょう。</p>
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