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<title>存储最多程序问题</title>
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<h1 id="存储最多程序问题">存储最多程序问题</h1>
<hr>
<p><strong>实例：</strong> <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 个程序，其存储所需容量分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>L</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>L</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">L_{1}, L_{2}, \cdots, L_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;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">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;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">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;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 mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 。两个磁盘，其容量都 为
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span></span></span></span>。</p>
<p><strong>询问：</strong> 若不允许一个程序同时存放于两个磁盘内，则这两个磁盘最多能存储多少个程序?</p>
<hr>
<p>该问题是一个 NP-hard 问题，可以通过 <a href="../doc/par.html">划分问题</a> 图灵归约证明</p>
<h2 id="np-hard-证明">NP-hard 证明</h2>
<p>假设存在一个求解该问题的算法 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>L</mi><mn>1</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>L</mi><mi>n</mi></msub><mo separator="true">;</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(L_1,...,L_n;L)</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 mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">L</span><span class="mclose">)</span></span></span></span>，对于一般划分问题实例 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>S</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A, S(a)</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 mathdefault">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mclose">)</span></span></span></span>，输入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo separator="true">;</mo><mi>S</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(a_1,...,a_n;S(A)/2)</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 mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathdefault">A</span><span class="mclose">)</span><span class="mord">/</span><span class="mord">2</span><span class="mclose">)</span></span></span></span>，如果有解，则划分问题输出 YES，否则划分问题输出 NO，得证。</p>
<h2 id="近似算法性能比证明">近似算法性能比证明</h2>
<hr>
<p><strong>算法：求解存储最多程序问题</strong></p>
<ol>
<li>对 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 个程序, 按其存储所需容理的非降顺序编号, 即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>⩽</mo><msub><mi>L</mi><mn>2</mn></msub><mo>⩽</mo><mo>⋯</mo><mo>⩽</mo><msub><mi>L</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">L_{1} \leqslant L_{2} \leqslant \cdots \leqslant L_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;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">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">⩽</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;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">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">⩽</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7733399999999999em;vertical-align:-0.13667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">⩽</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;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 mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</li>
<li>按程序编号从 1 到 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 依次向磁盘 1 存放程序，磁盘 1 放不下时， 转向磁盘 2 存放程序。</li>
<li>依次按照程序顺序向磁盘 2 存放程序，到磁盘 2 放不下为止。</li>
<li>输出两个磁盘上的程序数。</li>
</ol>
<hr>
<p>即算法 <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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span> 为多项式时间绝对近似算法。</p>
<p>首先考虑一个假想的单磁盘，其容量为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>L</mi></mrow><annotation encoding="application/x-tex">2 L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault">L</span></span></span></span>，如果它能按照从小到大的顺序存储 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span> 个程序，那么易得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mi>P</mi><mi>T</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">OPT(I)\leq p</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 mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span>。</p>
<p>其次，注意到使用该算法在两个容量为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span></span></span></span> 的磁盘中装软件时，能够装的软件数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">A(I)\leq p</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 mathdefault">A</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span>，此时确认了算法能装的软件数量的上界，下面来确认下界。</p>
<p>假设按照算法 A ，在磁盘 1 中可以存放 j 个程序，那么第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">j+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>个程序，在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>L</mi></mrow><annotation encoding="application/x-tex">2L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault">L</span></span></span></span> 的磁盘中的装法和在两个<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span></span></span></span>的磁盘中的装法可能不同，如下图所示：</p>
<p><img src="fig/1.png" alt=""></p>
<p>如果第 j 个程序恰好填满了磁盘 1，或者空出来的空间多于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>L</mi></mrow><annotation encoding="application/x-tex">2L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault">L</span></span></span></span>磁盘装法最后剩下来的空间。那么算法 <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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span> 的解和在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>L</mi></mrow><annotation encoding="application/x-tex">2L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault">L</span></span></span></span> 的单磁盘中的解的情况相同，此时磁盘 2 中则能够装 下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>−</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">p-j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 个程序。</p>
<p>否则，因为程序大小按序号排列，所以磁盘 2 中一定会少装一个程序（且最多少装一个程序）。</p>
<p>因此确认了算法 A 解的下界，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>A</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p-1 \leq A(I)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">A</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span></span></span></span>。</p>
<p>又因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mi>P</mi><mi>T</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">OPT(I)\leq p</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 mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span></span></span></span>，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mi>P</mi><mi>T</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>A</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">OPT(I) \leq p \leq A(I)+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 mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">A</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>，即</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>O</mi><mi>P</mi><mi>T</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>−</mo><mi>A</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|OPT(I)-A(I)|\leq 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">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">A</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></p>
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