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algorithm/basic_algo/binary_search/index.html

@@ -1305,6 +1305,17 @@
 
 
 
+        <label class="md-nav__link md-nav__link--active" for="__toc">
+
+
+  <span class="md-ellipsis">
+    Binary Search
+  </span>
+
+
+          <span class="md-nav__icon md-icon"></span>
+        </label>
+
       <a href="./" class="md-nav__link md-nav__link--active">
 
 
@@ -1315,6 +1326,52 @@
 
       </a>
 
+
+
+<nav class="md-nav md-nav--secondary" aria-label="目錄">
+
+
+
+
+
+
+    <label class="md-nav__title" for="__toc">
+      <span class="md-nav__icon md-icon"></span>
+      目錄
+    </label>
+    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
+
+        <li class="md-nav__item">
+  <a href="#introduction" class="md-nav__link">
+    <span class="md-ellipsis">
+      Introduction
+    </span>
+  </a>
+
+</li>
+
+        <li class="md-nav__item">
+  <a href="#complexity" class="md-nav__link">
+    <span class="md-ellipsis">
+      Complexity
+    </span>
+  </a>
+
+</li>
+
+        <li class="md-nav__item">
+  <a href="#implementation" class="md-nav__link">
+    <span class="md-ellipsis">
+      Implementation
+    </span>
+  </a>
+
+</li>
+
+    </ul>
+
+</nav>
+
     </li>
 
 
@@ -1754,6 +1811,41 @@
 
 
 
+    <label class="md-nav__title" for="__toc">
+      <span class="md-nav__icon md-icon"></span>
+      目錄
+    </label>
+    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
+
+        <li class="md-nav__item">
+  <a href="#introduction" class="md-nav__link">
+    <span class="md-ellipsis">
+      Introduction
+    </span>
+  </a>
+
+</li>
+
+        <li class="md-nav__item">
+  <a href="#complexity" class="md-nav__link">
+    <span class="md-ellipsis">
+      Complexity
+    </span>
+  </a>
+
+</li>
+
+        <li class="md-nav__item">
+  <a href="#implementation" class="md-nav__link">
+    <span class="md-ellipsis">
+      Implementation
+    </span>
+  </a>
+
+</li>
+
+    </ul>
+
 </nav>
                   </div>
                 </div>
@@ -1784,6 +1876,98 @@
 
 
 <h1 id="binary_search">Binary Search<a class="headerlink" href="#binary_search" title="Permanent link">¶</a></h1>
+<h2 id="introduction">Introduction<a class="headerlink" href="#introduction" title="Permanent link">¶</a></h2>
+<p>Binary Search(二元搜尋法) 是<em>分治法(Divide and Conquer)</em>的一種應用，非常有效率，但只適用在<strong>單調(Monotonic)</strong>的序列或函數上，也就是說，必須是遞增或遞減的，或是你簡單理解成<em>排序過(sorted)</em>的，舉幾個例子:</p>
+<ol>
+<li><code>[1, 2, 3, 4, 5]</code></li>
+<li><code>['a', 'b', 'c', 'd', 'e']</code></li>
+<li><code>[32, 16, 8, 4, 2, 1]</code></li>
+<li><span class="arithmatex">\(f(x)=x^2, \ x \ge 0,\ [f(1), f(2), f(3), f(4), f(5)]\)</span></li>
+</ol>
+<p>函數 <span class="arithmatex">\(f(x)=x^2\)</span> 在 <span class="arithmatex">\(x \ge 0\)</span> 時是遞增的:</p>
+<p><img alt="" src="../media/binary_search_1.png" /></p>
+<div class="language-python highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"><a href="#__codelineno-0-1">1</a></span>
+<span class="normal"><a href="#__codelineno-0-2">2</a></span>
+<span class="normal"><a href="#__codelineno-0-3">3</a></span>
+<span class="normal"><a href="#__codelineno-0-4">4</a></span>
+<span class="normal"><a href="#__codelineno-0-5">5</a></span>
+<span class="normal"><a href="#__codelineno-0-6">6</a></span>
+<span class="normal"><a href="#__codelineno-0-7">7</a></span>
+<span class="normal"><a href="#__codelineno-0-8">8</a></span>
+<span class="normal"><a href="#__codelineno-0-9">9</a></span></pre></div></td><td class="code"><div><pre><span></span><code><span id="__span-0-1"><a id="__codelineno-0-1" name="__codelineno-0-1"></a><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+</span><span id="__span-0-2"><a id="__codelineno-0-2" name="__codelineno-0-2"></a>    <span class="k">return</span> <span class="n">x</span> <span class="o">**</span> <span class="mi">2</span>
+</span><span id="__span-0-3"><a id="__codelineno-0-3" name="__codelineno-0-3"></a>
+</span><span id="__span-0-4"><a id="__codelineno-0-4" name="__codelineno-0-4"></a>
+</span><span id="__span-0-5"><a id="__codelineno-0-5" name="__codelineno-0-5"></a><span class="n">xs</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
+</span><span id="__span-0-6"><a id="__codelineno-0-6" name="__codelineno-0-6"></a><span class="nb">print</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span>
+</span><span id="__span-0-7"><a id="__codelineno-0-7" name="__codelineno-0-7"></a><span class="c1"># [f(1), f(2), f(3), f(4), f(5)]</span>
+</span><span id="__span-0-8"><a id="__codelineno-0-8" name="__codelineno-0-8"></a><span class="n">ys</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">]</span>
+</span><span id="__span-0-9"><a id="__codelineno-0-9" name="__codelineno-0-9"></a><span class="nb">print</span><span class="p">(</span><span class="n">ys</span><span class="p">)</span>
+</span></code></pre></div></td></tr></table></div>
+<div class="language-text highlight"><table class="highlighttable"><tr><th colspan="2" class="filename"><span class="filename">Output</span></th></tr><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"><a href="#__codelineno-1-1">1</a></span>
+<span class="normal"><a href="#__codelineno-1-2">2</a></span></pre></div></td><td class="code"><div><pre><span></span><code><span id="__span-1-1"><a id="__codelineno-1-1" name="__codelineno-1-1"></a>[1, 2, 3, 4, 5]
+</span><span id="__span-1-2"><a id="__codelineno-1-2" name="__codelineno-1-2"></a>[1, 4, 9, 16, 25]
+</span></code></pre></div></td></tr></table></div>
+<p>最後一個例子是想跟你強調，<em>函數也可以用二元搜尋法</em>，例如想要找第一個 <span class="arithmatex">\(f(x) \ge 10\)</span> 的整數 <span class="arithmatex">\(x\)</span>，我們可以用二元搜尋法找到 <span class="arithmatex">\(x=4\)</span>。</p>
+<h2 id="complexity">Complexity<a class="headerlink" href="#complexity" title="Permanent link">¶</a></h2>
+<p>玩過猜數字嗎?你每猜一次，我會告訴你猜的數字是太大還是太小，讓你從 <span class="arithmatex">\([1, 100]\)</span> 間猜一個數字，怎麼猜會最有效率?如果你第一次就猜 <span class="arithmatex">\(50\)</span>，我告訴你猜的數字太大，你就可以知道答案一定在 <span class="arithmatex">\([1, 49]\)</span> 之間，這樣你就可以<em>省下一半</em>的時間，你可以再猜 <span class="arithmatex">\(25\)</span>，而我告訴你猜的數字太小，那麼答案一定在 <span class="arithmatex">\([26, 49]\)</span> 之間，你又省下一半的時間，這就是二元搜尋法的精神。</p>
+<video muted controls autoplay loop>
+  <source src="../../basic_algo/media/binary_search_2.mp4" type="video/mp4">
+</video>
+
+<p>那最多要猜幾次呢?假設你真的很衰，猜到剩下 <span class="arithmatex">\([47, 48]\)</span> 這兩個數字在選的時候又猜錯，你猜了 <span class="arithmatex">\(47\)</span> 而我告訴你猜太小，那就只剩下 <span class="arithmatex">\(48\)</span> 了，也就是區間 <span class="arithmatex">\([48, 48]\)</span> <strong>剩下一個元素，這是我們的終止條件。</strong></p>
+<p>考慮一般的情況，有 <span class="arithmatex">\(n\)</span> 個數字，每次猜都可以把區間縮小一半，設我們要猜 <span class="arithmatex">\(k\)</span> 次才能把區間縮小到剩下一個元素:</p>
+<div class="arithmatex">\[
+\frac{n}{2^k} = 1
+\]</div>
+<div class="arithmatex">\[
+\Rightarrow n = 2^k
+\]</div>
+<div class="arithmatex">\[
+\Rightarrow \log_2 n = k
+\]</div>
+<p>那麼時間複雜度就是 <span class="arithmatex">\(\mathcal{O}(k) = \mathcal{O}(\log n)\)</span>。</p>
+<h2 id="implementation">Implementation<a class="headerlink" href="#implementation" title="Permanent link">¶</a></h2>
+<p>在寫程式之前，先來看動畫，有一個已排序過的串列 <code>[1, 1, 4, 5, 7, 8, 9, 10, 11, 12]</code>，我們要找 <code>4</code>:</p>
+<video muted controls autoplay loop>
+  <source src="../../basic_algo/media/binary_search_3.mp4" type="video/mp4">
+</video>
+
+<p>再來看程式碼:</p>
+<div class="language-python highlight"><table class="highlighttable"><tr><th colspan="2" class="filename"><span class="filename">binary_search.py</span></th></tr><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"><a href="#__codelineno-2-1"> 1</a></span>
+<span class="normal"><a href="#__codelineno-2-2"> 2</a></span>
+<span class="normal"><a href="#__codelineno-2-3"> 3</a></span>
+<span class="normal"><a href="#__codelineno-2-4"> 4</a></span>
+<span class="normal"><a href="#__codelineno-2-5"> 5</a></span>
+<span class="normal"><a href="#__codelineno-2-6"> 6</a></span>
+<span class="normal"><a href="#__codelineno-2-7"> 7</a></span>
+<span class="normal"><a href="#__codelineno-2-8"> 8</a></span>
+<span class="normal"><a href="#__codelineno-2-9"> 9</a></span>
+<span class="normal"><a href="#__codelineno-2-10">10</a></span>
+<span class="normal"><a href="#__codelineno-2-11">11</a></span>
+<span class="normal"><a href="#__codelineno-2-12">12</a></span>
+<span class="normal"><a href="#__codelineno-2-13">13</a></span>
+<span class="normal"><a href="#__codelineno-2-14">14</a></span>
+<span class="normal"><a href="#__codelineno-2-15">15</a></span>
+<span class="normal"><a href="#__codelineno-2-16">16</a></span></pre></div></td><td class="code"><div><pre><span></span><code><span id="__span-2-1"><a id="__codelineno-2-1" name="__codelineno-2-1"></a><span class="k">def</span> <span class="nf">binary_search</span><span class="p">(</span><span class="n">lst</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
+</span><span id="__span-2-2"><a id="__codelineno-2-2" name="__codelineno-2-2"></a>    <span class="n">left</span><span class="p">,</span> <span class="n">right</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">lst</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+</span><span id="__span-2-3"><a id="__codelineno-2-3" name="__codelineno-2-3"></a>
+</span><span id="__span-2-4"><a id="__codelineno-2-4" name="__codelineno-2-4"></a>    <span class="k">while</span> <span class="n">left</span> <span class="o">&lt;=</span> <span class="n">right</span><span class="p">:</span>
+</span><span id="__span-2-5"><a id="__codelineno-2-5" name="__codelineno-2-5"></a>        <span class="n">mid</span> <span class="o">=</span> <span class="p">(</span><span class="n">left</span> <span class="o">+</span> <span class="n">right</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
+</span><span id="__span-2-6"><a id="__codelineno-2-6" name="__codelineno-2-6"></a>        <span class="k">if</span> <span class="n">lst</span><span class="p">[</span><span class="n">mid</span><span class="p">]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
+</span><span id="__span-2-7"><a id="__codelineno-2-7" name="__codelineno-2-7"></a>            <span class="k">return</span> <span class="n">mid</span>
+</span><span id="__span-2-8"><a id="__codelineno-2-8" name="__codelineno-2-8"></a>        <span class="k">elif</span> <span class="n">lst</span><span class="p">[</span><span class="n">mid</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">target</span><span class="p">:</span>
+</span><span id="__span-2-9"><a id="__codelineno-2-9" name="__codelineno-2-9"></a>            <span class="n">left</span> <span class="o">=</span> <span class="n">mid</span> <span class="o">+</span> <span class="mi">1</span>
+</span><span id="__span-2-10"><a id="__codelineno-2-10" name="__codelineno-2-10"></a>        <span class="k">else</span><span class="p">:</span>
+</span><span id="__span-2-11"><a id="__codelineno-2-11" name="__codelineno-2-11"></a>            <span class="n">right</span> <span class="o">=</span> <span class="n">mid</span> <span class="o">-</span> <span class="mi">1</span>
+</span><span id="__span-2-12"><a id="__codelineno-2-12" name="__codelineno-2-12"></a>
+</span><span id="__span-2-13"><a id="__codelineno-2-13" name="__codelineno-2-13"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
+</span><span id="__span-2-14"><a id="__codelineno-2-14" name="__codelineno-2-14"></a>
+</span><span id="__span-2-15"><a id="__codelineno-2-15" name="__codelineno-2-15"></a><span class="n">nums</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]</span>
+</span><span id="__span-2-16"><a id="__codelineno-2-16" name="__codelineno-2-16"></a><span class="nb">print</span><span class="p">(</span><span class="n">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+</span></code></pre></div></td></tr></table></div>
+<div class="language-text highlight"><table class="highlighttable"><tr><th colspan="2" class="filename"><span class="filename">Output</span></th></tr><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"><a href="#__codelineno-3-1">1</a></span></pre></div></td><td class="code"><div><pre><span></span><code><span id="__span-3-1"><a id="__codelineno-3-1" name="__codelineno-3-1"></a>2
+</span></code></pre></div></td></tr></table></div>
 
 
 

algorithm/basic_algo/linear_search/index.html

@@ -1916,7 +1916,7 @@ <h2 id="introduction">Introduction<a class="headerlink" href="#introduction" tit
 <p>這應該是最簡單的演算法之一，不過我們可以來複習上一章所學的時間複雜度。</p>
 <h2 id="implementation">Implementation<a class="headerlink" href="#implementation" title="Permanent link">¶</a></h2>
 <p>直接來看程式碼，定義 <code>linear_search(lst, target)</code> 函式，這個函式接受一個串列 <code>lst</code> 和一個目標值 <code>target</code>，回傳目標值在串列中的索引，如果目標值不存在就回傳 <code>-1</code>。</p>
-<div class="language-python highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"><a href="#__codelineno-0-1"> 1</a></span>
+<div class="language-python highlight"><table class="highlighttable"><tr><th colspan="2" class="filename"><span class="filename">linear_search.py</span></th></tr><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"><a href="#__codelineno-0-1"> 1</a></span>
 <span class="normal"><a href="#__codelineno-0-2"> 2</a></span>
 <span class="normal"><a href="#__codelineno-0-3"> 3</a></span>
 <span class="normal"><a href="#__codelineno-0-4"> 4</a></span>