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data-v-4c05ee02><meta property="position" content="4" data-v-4c05ee02></li><!--]--></ol></nav><!--[--><!--]--><!--[--><div class="vp-doc-title" data-v-9ebb517f><!--[--><!--]--><h1 class="page-title" data-v-9ebb517f><!----> 快速幂算法详解 <!----></h1><!--[--><!--]--></div><div class="vp-doc-meta" data-v-9ebb517f><!--[--><!--]--><p class="reading-time" data-v-9ebb517f><span class="vpi-books icon" data-v-9ebb517f></span><span data-v-9ebb517f>约 555 字</span><span data-v-9ebb517f>大约 2 分钟</span></p><p data-v-9ebb517f><span class="vpi-tag icon" data-v-9ebb517f></span><!--[--><a class="vp-link link tag vp-tag-mpia" href="/article/tags/?tag=algorithms" data-v-9ebb517f><!--[-->algorithms<!--]--></a><!--]--></p><!--[--><!--]--><p class="create-time" data-v-9ebb517f><span class="vpi-clock icon" data-v-9ebb517f></span><span data-v-9ebb517f>2026-01-09</span></p></div><!--]--><!--[--><!--]--><!--[--><div class="_archives_1325a3bf-91d7-43ff-9630-e894549e12c1_ external-link-icon-enabled vp-doc plume-content" vp-content data-v-a4311ef5><!--[--><!--]--><div data-v-a4311ef5><h2 id="简介" tabindex="-1"><a class="header-anchor" href="#简介"><span>简介</span></a></h2><p>快速幂算法是用于快速计算的算法，可以用于快速的处理大整数幂的场景。</p><p>最简单的 for 循环求数字的 n 次幂需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的时间复杂度。快速幂方法可以达到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mclose">)</span></span></span></span> 的时间复杂度。</p><p>快速幂的核心思想是将指数拆分，以达到快速计算的目的。</p><h2 id="快速幂-二进制法" tabindex="-1"><a class="header-anchor" href="#快速幂-二进制法"><span>快速幂 - 二进制法</span></a></h2><h3 id="原理" tabindex="-1"><a class="header-anchor" href="#原理"><span>原理</span></a></h3><p>二进制法的核心思想是将指数转换为二进制形式，通过逐位处理并结合平方运算减少乘法次数。</p><p><strong>二进制分解指数</strong> 将指数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 表示为二进制形式，例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>13</mn></mrow><annotation encoding="application/x-tex">n=13</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">13</span></span></span></span> 对应二进制为 <code>1101</code>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>13</mn><mo>=</mo><mn>8</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">13=8+4+1=2^3+2^2+2^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">13</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span>。</p><p><strong>幂的拆分</strong> 根据二进制分解，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>a</mi><msup><mn>2</mn><mi>k</mi></msup></msup><mo>×</mo><msup><mi>a</mi><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></msup><mo>×</mo><mo>⋯</mo><mo>×</mo><msup><mi>a</mi><msup><mn>2</mn><mn>0</mn></msup></msup></mrow><annotation encoding="application/x-tex">a^n=a^{2^k} \times a^{2^{k-1}} \times \dots \times a^{2^0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0952em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0119em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.927em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0952em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0119em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.927em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9869em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>，其中仅当二进制位为 1 时，对应项被保留。</p><p><strong>逐位处理与平方加速</strong></p><ul><li>从最低位到最高位依次处理二进制每一位。</li><li>若当前位为 1，则将当前的底数累乘到结果中。</li><li>每一步将底数平方，为处理更高位做准备。</li></ul><h3 id="代码示例" tabindex="-1"><a class="header-anchor" href="#代码示例"><span>代码示例</span></a></h3><div class="language-python line-numbers-mode" data-highlighter="shiki" data-ext="python" style="--shiki-light:#393a34;--shiki-dark:#dbd7caee;--shiki-light-bg:#ffffff;--shiki-dark-bg:#121212;"><pre class="shiki shiki-themes vitesse-light vitesse-dark vp-code"><code class="language-python"><span class="line"><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">def</span><span style="--shiki-light:#59873A;--shiki-dark:#80A665;"> power</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">(</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;">base</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">,</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">):</span></span>
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<span class="line"><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> res </span><span style="--shiki-light:#999999;--shiki-dark:#666666;">=</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 1</span></span>
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<span class="line"><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> while</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">></span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 0</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">:</span></span>
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<span class="line"><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> if</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">&</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 1</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">:</span></span>
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<span class="line"><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> res </span><span style="--shiki-light:#999999;--shiki-dark:#666666;">*=</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> base</span></span>
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<span class="line"><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> base </span><span style="--shiki-light:#999999;--shiki-dark:#666666;">*=</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> base</span></span>
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<span class="line"><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp </span><span style="--shiki-light:#999999;--shiki-dark:#666666;">>>=</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 1</span></span>
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<span class="line"><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> return</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> res</span></span></code></pre><div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0;"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div><h2 id="快速幂-折半法" tabindex="-1"><a class="header-anchor" href="#快速幂-折半法"><span>快速幂 - 折半法</span></a></h2><h3 id="原理-1" tabindex="-1"><a class="header-anchor" href="#原理-1"><span>原理</span></a></h3><p>折半法的核心公式如下：</p><p>我们要快速计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次方：</p><ul><li>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 为奇数的时候：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^n = a \cdot (a^2)^{(n-1)/2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span></span></span></span></li><li>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 为偶数的时候：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^n = (a^2)^{n/2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord mtight">/2</span></span></span></span></span></span></span></span></span></span></span></span></li><li>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 为 0 的时候，直接返回 1</li></ul><p>通过递归或迭代，将大指数问题分解为小指数问题，最后合并结果。</p><h3 id="代码示例-1" tabindex="-1"><a class="header-anchor" href="#代码示例-1"><span>代码示例</span></a></h3><div class="language-python line-numbers-mode" data-highlighter="shiki" data-ext="python" style="--shiki-light:#393a34;--shiki-dark:#dbd7caee;--shiki-light-bg:#ffffff;--shiki-dark-bg:#121212;"><pre class="shiki shiki-themes vitesse-light vitesse-dark vp-code"><code class="language-python"><span class="line"><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">def</span><span style="--shiki-light:#59873A;--shiki-dark:#80A665;"> power</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">(</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;">base</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">,</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">):</span></span>
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<span class="line"><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> if</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">==</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 0</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">:</span><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> return</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 1</span></span>
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<span class="line"><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> half </span><span style="--shiki-light:#999999;--shiki-dark:#666666;">=</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> power</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">(</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;">base</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">,</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">//</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 2</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">)</span></span>
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<span class="line"><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> return</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> half </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">*</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> half </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">*</span><span style="--shiki-light:#999999;--shiki-dark:#666666;"> (</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;">base </span><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;">if</span><span style="--shiki-light:#393A34;--shiki-dark:#DBD7CAEE;"> exp </span><span style="--shiki-light:#AB5959;--shiki-dark:#CB7676;">%</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 2</span><span style="--shiki-light:#1E754F;--shiki-dark:#4D9375;"> else</span><span style="--shiki-light:#2F798A;--shiki-dark:#4C9A91;"> 1</span><span style="--shiki-light:#999999;--shiki-dark:#666666;">)</span></span></code></pre><div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0;"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div><h2 id="两种方法对比" tabindex="-1"><a class="header-anchor" href="#两种方法对比"><span>两种方法对比</span></a></h2><table><thead><tr><th style="text-align:left;">特性</th><th style="text-align:left;">二进制法</th><th style="text-align:left;">折半法</th></tr></thead><tbody><tr><td style="text-align:left;"><strong>实现方式</strong></td><td style="text-align:left;">迭代 + 位运算</td><td style="text-align:left;">递归/迭代 + 分治</td></tr><tr><td style="text-align:left;"><strong>时间复杂度</strong></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mclose">)</span></span></span></span></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mclose">)</span></span></span></span></td></tr><tr><td style="text-align:left;"><strong>空间复杂度</strong></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mclose">)</span></span></span></span> （递归）</td></tr></tbody></table><p>两者都可以实现在指数时间解决问题，二进制方法比折半法更加的省空间。</p></div><!----><!----><!----><footer class="vp-doc-footer" data-v-a4311ef5 data-v-abf7cea9><!--[--><!--]--><!----><div class="contributors" aria-label="Contributors" data-v-abf7cea9><span class="contributors-label" data-v-abf7cea9>贡献者: </span><span class="contributors-info" data-v-abf7cea9><!--[--><!--[--><span class="contributor" data-v-abf7cea9>祀梦</span><!----><!--]--><!--]--></span></div><nav class="prev-next" data-v-abf7cea9><div class="pager" data-v-abf7cea9><a class="vp-link link pager-link prev" href="/archives/ea20bdda-0d49-4472-a647-2e305a930d11/" data-v-abf7cea9><!--[--><span class="desc" data-v-abf7cea9>上一页</span><span class="title" data-v-abf7cea9><!----><span data-v-abf7cea9>通过位运算快速生成所有的子序列</span></span><!--]--></a></div><div class="pager" data-v-abf7cea9><a class="vp-link link pager-link next" href="/archives/b1c77a1d-d402-4788-8049-fa3aeb12ebd0/" data-v-abf7cea9><!--[--><span class="desc" 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