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第 0 章   はじめに
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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
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1.3   まとめ
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第 2 章   計算量の解析
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第 2 章   計算量の解析
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2.1   アルゴリズムの効率の評価
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2.2   反復と再帰
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2.3   時間計算量
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2.4   空間計算量
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2.5   まとめ
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2.6   練習問題
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第 3 章   データ構造
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第 3 章   データ構造
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3.1   データ構造の分類
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3.2   基本データ型
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3.3   数値の符号化 *
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3.4   文字エンコーディング *
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3.5   まとめ
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3.6   練習問題
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第 4 章   配列と連結リスト
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第 4 章   配列と連結リスト
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4.1   配列
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4.2   連結リスト
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4.3   リスト
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4.4   メモリとキャッシュ *
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4.5   まとめ
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4.6   練習問題
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第 5 章   スタックとキュー
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第 5 章   スタックとキュー
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5.1   スタック
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5.2   キュー
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5.3   両端キュー
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5.4   まとめ
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5.5   練習問題
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第 6 章   ハッシュテーブル
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第 6 章   ハッシュテーブル
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6.1   ハッシュテーブル
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6.2   ハッシュ衝突
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6.3   ハッシュアルゴリズム
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6.4   まとめ
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6.5   練習問題
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第 7 章   木
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第 7 章   木
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7.1   二分木
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7.2   二分木の走査
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7.3   二分木の配列表現
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7.4   二分探索木
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7.5   AVL 木 *
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7.6   まとめ
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7.7   練習問題
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第 8 章   ヒープ
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第 8 章   ヒープ
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8.1   ヒープ
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8.2   ヒープの構築
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8.3   Top-k 問題
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8.4   まとめ
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8.5   練習問題
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第 9 章   グラフ
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第 9 章   グラフ
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9.1   グラフ
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9.2   グラフの基本操作
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9.3   グラフの走査
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9.4   まとめ
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9.4   まとめ
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1.   要点の振り返り
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2.   Q &amp; A
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9.5   練習問題
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第 10 章   探索
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第 10 章   探索
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10.1   二分探索
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10.2   二分探索で挿入位置を探す
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10.3   二分探索で境界を探す
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10.4   ハッシュを使った最適化
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10.5   探索アルゴリズムを改めて考える
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10.6   まとめ
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10.7   練習問題
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第 11 章   ソート
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第 11 章   ソート
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11.1   ソートアルゴリズム
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11.2   選択ソート
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11.3   バブルソート
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11.4   挿入ソート
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11.5   クイックソート
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11.6   マージソート
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11.7   ヒープソート
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11.8   バケットソート
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11.9   カウントソート
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11.10   基数ソート
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11.11   まとめ
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11.12   練習問題
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第 12 章   分割統治
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第 12 章   分割統治
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12.1   分割統治アルゴリズム
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12.2   分割統治による探索戦略
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12.3   二分木の構築問題
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12.4   ハノイの塔の問題
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12.5   まとめ
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12.6   練習問題
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第 13 章   バックトラッキング
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第 13 章   バックトラッキング
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13.1   バックトラッキングアルゴリズム
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13.2   順列列挙問題
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13.3   部分集合和問題
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13.4   n クイーン問題
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13.5   まとめ
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13.6   練習問題
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第 14 章   動的計画法
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14.1   動的計画法への第一歩
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14.2   動的計画法の問題の特徴
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14.3   動的計画法の解き方
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14.4   0-1 ナップサック問題
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14.5   個数制限なしナップサック問題
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14.6   編集距離問題
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14.7   まとめ
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14.8   練習問題
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第 15 章   貪欲法
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<h1 id="94">9.4   まとめ<a class="headerlink" href="#94" title="この見出しへのリンク"></a></h1>
<h3 id="1">1.   要点の振り返り<a class="headerlink" href="#1" title="この見出しへのリンク"></a></h3>
<ul>
<li>グラフは頂点と辺で構成され、頂点の集合と辺の集合をまとめたものとして表せます。</li>
<li>線形の関係(連結リスト)や分割統治の関係(木)に比べ、ネットワークの関係(グラフ)は自由度が高く、その分複雑です。</li>
<li>有向グラフの辺には向きがあります。連結グラフではどの頂点にも到達でき、重み付きグラフでは各辺に重みの変数があります。</li>
<li>隣接行列は行列でグラフを表します。各行(列)が一つの頂点、行列の要素が辺に対応し、<span class="arithmatex">\(1\)</span> または <span class="arithmatex">\(0\)</span> で 2 頂点間に辺があるかどうかを示します。辺の追加・削除・検索・更新は効率的ですが、多くの空間を使います。</li>
<li>隣接リストは複数の連結リストでグラフを表します。<span class="arithmatex">\(i\)</span> 番目の連結リストは頂点 <span class="arithmatex">\(i\)</span> に対応し、その頂点に隣接するすべての頂点を格納します。隣接行列より空間を節約できますが、辺を探すには連結リストを走査する必要があるため、時間効率は低くなります。</li>
<li>隣接リスト内の連結リストが長くなりすぎた場合は、赤黒木やハッシュテーブルに変換して探索効率を高められます。</li>
<li>アルゴリズムの考え方として、隣接行列は「空間を使って時間を節約する」、隣接リストは「時間を使って空間を節約する」方法です。</li>
<li>グラフは、ソーシャルネットワークや地下鉄の路線など、さまざまな現実のシステムのモデル化に使えます。</li>
<li>木はグラフの特殊な場合であり、木の走査もグラフの走査の特殊な場合です。</li>
<li>グラフの幅優先走査は、近いところから遠いところへ、層ごとに広がる探索方法で、通常はキューを使って実装します。</li>
<li>グラフの深さ優先走査は、まず進めるところまで進み、進めなくなったら引き返す探索方法で、通常は再帰で実装します。</li>
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<h3 id="2-q-a">2.   Q &amp; A<a class="headerlink" href="#2-q-a" title="この見出しへのリンク"></a></h3>
<p><strong>Q</strong>:経路は頂点の列ですか、それとも辺の列ですか。</p>
<p>Wikipedia では言語版によって定義が異なります。英語版では「経路は辺の列」、中国語版では「経路は頂点の列」とされています。英語版の原文は次のとおりです。In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p>
<p>本書では、経路を頂点の列ではなく、辺の列として扱います。2 頂点間を複数の辺が結ぶことがあり、その場合はそれぞれの辺が一つの経路に対応するためです。</p>
<p><strong>Q</strong>:非連結グラフには、走査で到達できない頂点がありますか。</p>
<p>非連結グラフでは、ある頂点から出発したとき、到達できない頂点が少なくとも一つあります。すべての連結成分を走査するには、複数の開始点を設定する必要があります。</p>
<p><strong>Q</strong>:隣接リストにある「その頂点につながるすべての頂点」の順序には、決まりがありますか。</p>
<p>順序は任意です。ただし、実際の用途では、頂点を追加した順や頂点の値の大小順など、特定の規則に従って並べることがあります。これにより、「何らかの値が最大または最小である頂点」をすばやく探しやすくなります。</p>
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