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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">bisect</span></code> --- 数组二分查找算法</a><ul>
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<section id="module-bisect">
<span id="bisect-array-bisection-algorithm"></span><h1><a class="reference internal" href="#module-bisect" title="bisect: Array bisection algorithms for binary searching."><code class="xref py py-mod docutils literal notranslate"><span class="pre">bisect</span></code></a> --- 数组二分查找算法<a class="headerlink" href="#module-bisect" title="永久链接至标题"></a></h1>
<p><strong>源代码：</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.8/Lib/bisect.py">Lib/bisect.py</a></p>
<hr class="docutils" />
<p>这个模块对有序列表提供了支持，使得他们可以在插入新数据仍然保持有序。对于长列表，如果其包含元素的比较操作十分昂贵的话，这可以是对更常见方法的改进。这个模块叫做 <a class="reference internal" href="#module-bisect" title="bisect: Array bisection algorithms for binary searching."><code class="xref py py-mod docutils literal notranslate"><span class="pre">bisect</span></code></a> 因为其使用了基本的二分（bisection）算法。源代码也可以作为很棒的算法示例（边界判断也做好啦！）</p>
<p>定义了以下函数：</p>
<dl class="function">
<dt id="bisect.bisect_left">
<code class="sig-prename descclassname">bisect.</code><code class="sig-name descname">bisect_left</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">x</em>, <em class="sig-param">lo=0</em>, <em class="sig-param">hi=len(a)</em><span class="sig-paren">)</span><a class="headerlink" href="#bisect.bisect_left" title="永久链接至目标"></a></dt>
<dd><p>在 <em>a</em> 中找到 <em>x</em> 合适的插入点以维持有序。参数 <em>lo</em> 和 <em>hi</em> 可以被用于确定需要考虑的子集；默认情况下整个列表都会被使用。如果 <em>x</em> 已经在 <em>a</em> 里存在，那么插入点会在已存在元素之前（也就是左边）。如果 <em>a</em> 是列表（list）的话，返回值是可以被放在 <code class="docutils literal notranslate"><span class="pre">list.insert()</span></code> 的第一个参数的。</p>
<p>返回的插入点 <em>i</em> 可以将数组 <em>a</em> 分成两部分。左侧是 <code class="docutils literal notranslate"><span class="pre">all(val</span> <span class="pre">&lt;</span> <span class="pre">x</span> <span class="pre">for</span> <span class="pre">val</span> <span class="pre">in</span> <span class="pre">a[lo:i])</span></code> ，右侧是 <code class="docutils literal notranslate"><span class="pre">all(val</span> <span class="pre">&gt;=</span> <span class="pre">x</span> <span class="pre">for</span> <span class="pre">val</span> <span class="pre">in</span> <span class="pre">a[i:hi])</span></code> 。</p>
</dd></dl>
<dl class="function">
<dt id="bisect.bisect_right">
<code class="sig-prename descclassname">bisect.</code><code class="sig-name descname">bisect_right</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">x</em>, <em class="sig-param">lo=0</em>, <em class="sig-param">hi=len(a)</em><span class="sig-paren">)</span><a class="headerlink" href="#bisect.bisect_right" title="永久链接至目标"></a></dt>
<dt id="bisect.bisect">
<code class="sig-prename descclassname">bisect.</code><code class="sig-name descname">bisect</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">x</em>, <em class="sig-param">lo=0</em>, <em class="sig-param">hi=len(a)</em><span class="sig-paren">)</span><a class="headerlink" href="#bisect.bisect" title="永久链接至目标"></a></dt>
<dd><p>类似于 <a class="reference internal" href="#bisect.bisect_left" title="bisect.bisect_left"><code class="xref py py-func docutils literal notranslate"><span class="pre">bisect_left()</span></code></a>，但是返回的插入点是 <em>a</em> 中已存在元素 <em>x</em> 的右侧。</p>
<p>返回的插入点 <em>i</em> 可以将数组 <em>a</em> 分成两部分。左侧是 <code class="docutils literal notranslate"><span class="pre">all(val</span> <span class="pre">&lt;=</span> <span class="pre">x</span> <span class="pre">for</span> <span class="pre">val</span> <span class="pre">in</span> <span class="pre">a[lo:i])</span></code>，右侧是 <code class="docutils literal notranslate"><span class="pre">all(val</span> <span class="pre">&gt;</span> <span class="pre">x</span> <span class="pre">for</span> <span class="pre">val</span> <span class="pre">in</span> <span class="pre">a[i:hi])</span></code> for the right side。</p>
</dd></dl>
<dl class="function">
<dt id="bisect.insort_left">
<code class="sig-prename descclassname">bisect.</code><code class="sig-name descname">insort_left</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">x</em>, <em class="sig-param">lo=0</em>, <em class="sig-param">hi=len(a)</em><span class="sig-paren">)</span><a class="headerlink" href="#bisect.insort_left" title="永久链接至目标"></a></dt>
<dd><p>将 <em>x</em> 插入到一个有序序列 <em>a</em> 里，并维持其有序。如果 <em>a</em> 有序的话，这相当于 <code class="docutils literal notranslate"><span class="pre">a.insert(bisect.bisect_left(a,</span> <span class="pre">x,</span> <span class="pre">lo,</span> <span class="pre">hi),</span> <span class="pre">x)</span></code>。要注意搜索是 O(log n) 的，插入却是 O(n) 的。</p>
</dd></dl>
<dl class="function">
<dt id="bisect.insort_right">
<code class="sig-prename descclassname">bisect.</code><code class="sig-name descname">insort_right</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">x</em>, <em class="sig-param">lo=0</em>, <em class="sig-param">hi=len(a)</em><span class="sig-paren">)</span><a class="headerlink" href="#bisect.insort_right" title="永久链接至目标"></a></dt>
<dt id="bisect.insort">
<code class="sig-prename descclassname">bisect.</code><code class="sig-name descname">insort</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">x</em>, <em class="sig-param">lo=0</em>, <em class="sig-param">hi=len(a)</em><span class="sig-paren">)</span><a class="headerlink" href="#bisect.insort" title="永久链接至目标"></a></dt>
<dd><p>类似于 <a class="reference internal" href="#bisect.insort_left" title="bisect.insort_left"><code class="xref py py-func docutils literal notranslate"><span class="pre">insort_left()</span></code></a>，但是把 <em>x</em> 插入到 <em>a</em> 中已存在元素 <em>x</em> 的右侧。</p>
</dd></dl>
<div class="admonition seealso">
<p class="admonition-title">参见</p>
<p><a class="reference external" href="https://code.activestate.com/recipes/577197-sortedcollection/">SortedCollection recipe</a> 使用 bisect 构造了一个功能完整的集合类，提供了直接的搜索方法和对用于搜索的 key 方法的支持。所有用于搜索的键都是预先计算的，以避免在搜索时对 key 方法的不必要调用。</p>
</div>
<section id="searching-sorted-lists">
<h2>搜索有序列表<a class="headerlink" href="#searching-sorted-lists" title="永久链接至标题"></a></h2>
<p>上面的 <a class="reference internal" href="#module-bisect" title="bisect: Array bisection algorithms for binary searching."><code class="xref py py-func docutils literal notranslate"><span class="pre">bisect()</span></code></a> 函数对于找到插入点是有用的，但在一般的搜索任务中可能会有点尴尬。下面 5 个函数展示了如何将其转变成有序列表中的标准查找函数</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="s1">&#39;Locate the leftmost value exactly equal to x&#39;</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">bisect_left</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="ow">and</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
<span class="k">return</span> <span class="n">i</span>
<span class="k">raise</span> <span class="ne">ValueError</span>
<span class="k">def</span> <span class="nf">find_lt</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="s1">&#39;Find rightmost value less than x&#39;</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">bisect_left</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span><span class="p">:</span>
<span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">raise</span> <span class="ne">ValueError</span>
<span class="k">def</span> <span class="nf">find_le</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="s1">&#39;Find rightmost value less than or equal to x&#39;</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">bisect_right</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span><span class="p">:</span>
<span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">raise</span> <span class="ne">ValueError</span>
<span class="k">def</span> <span class="nf">find_gt</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="s1">&#39;Find leftmost value greater than x&#39;</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">bisect_right</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
<span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">raise</span> <span class="ne">ValueError</span>
<span class="k">def</span> <span class="nf">find_ge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="s1">&#39;Find leftmost item greater than or equal to x&#39;</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">bisect_left</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
<span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">raise</span> <span class="ne">ValueError</span>
</pre></div>
</div>
</section>
<section id="other-examples">
<h2>其他示例<a class="headerlink" href="#other-examples" title="永久链接至标题"></a></h2>
<p id="bisect-example">函数 <a class="reference internal" href="#module-bisect" title="bisect: Array bisection algorithms for binary searching."><code class="xref py py-func docutils literal notranslate"><span class="pre">bisect()</span></code></a> 还可以用于数字表查询。这个例子是使用 <a class="reference internal" href="#module-bisect" title="bisect: Array bisection algorithms for binary searching."><code class="xref py py-func docutils literal notranslate"><span class="pre">bisect()</span></code></a> 从一个给定的考试成绩集合里，通过一个有序数字表，查出其对应的字母等级：90 分及以上是 'A'，80 到 89 是 'B'，以此类推</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">grade</span><span class="p">(</span><span class="n">score</span><span class="p">,</span> <span class="n">breakpoints</span><span class="o">=</span><span class="p">[</span><span class="mi">60</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">80</span><span class="p">,</span> <span class="mi">90</span><span class="p">],</span> <span class="n">grades</span><span class="o">=</span><span class="s1">&#39;FDCBA&#39;</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">i</span> <span class="o">=</span> <span class="n">bisect</span><span class="p">(</span><span class="n">breakpoints</span><span class="p">,</span> <span class="n">score</span><span class="p">)</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">grades</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="gp">...</span>
<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">grade</span><span class="p">(</span><span class="n">score</span><span class="p">)</span> <span class="k">for</span> <span class="n">score</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">33</span><span class="p">,</span> <span class="mi">99</span><span class="p">,</span> <span class="mi">77</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">90</span><span class="p">,</span> <span class="mi">100</span><span class="p">]]</span>
<span class="go">[&#39;F&#39;, &#39;A&#39;, &#39;C&#39;, &#39;C&#39;, &#39;B&#39;, &#39;A&#39;, &#39;A&#39;]</span>
</pre></div>
</div>
<p>与 <a class="reference internal" href="functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate"><span class="pre">sorted()</span></code></a> 函数不同，对于 <a class="reference internal" href="#module-bisect" title="bisect: Array bisection algorithms for binary searching."><code class="xref py py-func docutils literal notranslate"><span class="pre">bisect()</span></code></a> 函数来说，<em>key</em> 或者 <em>reversed</em> 参数并没有什么意义。因为这会导致设计效率低下（连续调用 bisect 函数时，是不会 &quot;记住&quot; 过去查找过的键的）。</p>
<p>正相反，最好去搜索预先计算好的键列表，来查找相关记录的索引。</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">data</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;blue&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;yellow&#39;</span><span class="p">,</span> <span class="mi">8</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">keys</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">data</span><span class="p">]</span> <span class="c1"># precomputed list of keys</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span><span class="p">[</span><span class="n">bisect_left</span><span class="p">(</span><span class="n">keys</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span>
<span class="go">(&#39;black&#39;, 0)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span><span class="p">[</span><span class="n">bisect_left</span><span class="p">(</span><span class="n">keys</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
<span class="go">(&#39;blue&#39;, 1)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span><span class="p">[</span><span class="n">bisect_left</span><span class="p">(</span><span class="n">keys</span><span class="p">,</span> <span class="mi">5</span><span class="p">)]</span>
<span class="go">(&#39;red&#39;, 5)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">data</span><span class="p">[</span><span class="n">bisect_left</span><span class="p">(</span><span class="n">keys</span><span class="p">,</span> <span class="mi">8</span><span class="p">)]</span>
<span class="go">(&#39;yellow&#39;, 8)</span>
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