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<h1 class="post-title p-name">分治法</h1>
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<h5 id="基本思想"><a href="#基本思想" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>基本思想</h5>
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<p>将一个规模为n的问题分解为k个规模较小的子问题，这些子问题互相独立且与原问题相同。递归地解这些子问题，然后将各子问题的解合并得到原问题的解
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子问题数量不定，但最好使每个子问题的规模相等</p>
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<p>一般算法设计模式</p>
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<pre tabindex="0"><code>divide-and-conquer(p)
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{
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if(|p|<=n0)
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adhoc(p);
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divide p into smaller subinstances p1,p2,pk;
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for(i=1,i<=k,k++)
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yi=divide-and-conquer(pi);
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return merge(y1,···,yk);
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}
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</code></pre><hr>
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<h6 id="二分搜索技术"><a href="#二分搜索技术" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>二分搜索技术</h6>
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<p>将n个元素分成个数大致相同的两半，取a[n/2]与x进行比较。</p>
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<ol>
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<li>
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<p>$x=[n/2]$,则找到x，算法终止</p>
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</li>
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<li>
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<p>$x<a[n/2]$,在数组a的左半部继续搜索x</p>
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</li>
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<li>
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<p>$x>a[n/2]$,在数组a的右半部继续搜索x</p>
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<p>具体算法</p>
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<div class="highlight"><div class="chroma">
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<div class="table-container"><table class="lntable"><tr><td class="lntd">
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<pre tabindex="0" class="chroma"><code><span class="lnt"> 1
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</span><span class="lnt">15
|
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</span><span class="lnt">16
|
||
</span></code></pre></td>
|
||
<td class="lntd">
|
||
<pre tabindex="0" class="chroma"><code class="language-java" data-lang="java"><span class="line"><span class="cl"> <span class="kd">public</span> <span class="kd">static</span> <span class="kt">int</span> <span class="nf">binarySearch</span><span class="o">(</span><span class="kt">int</span> <span class="o">[]</span> <span class="n">a</span><span class="o">,</span><span class="kt">int</span> <span class="n">x</span><span class="o">,</span><span class="kt">int</span> <span class="n">n</span><span class="o">){</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//在a[0]<=a[1]<=···<=a[n-1]中搜索x
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="c1">//找到x时返回起在数组中的位置，则返回-1
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="kt">int</span> <span class="n">left</span> <span class="o">=</span><span class="mi">0</span><span class="o">;</span><span class="kt">int</span> <span class="n">right</span><span class="o">=</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">while</span><span class="o">(</span><span class="n">left</span><span class="o"><=</span><span class="n">right</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kt">int</span> <span class="n">middle</span><span class="o">=(</span><span class="n">left</span><span class="o">+</span><span class="n">right</span><span class="o">)/</span><span class="mi">2</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">if</span><span class="o">(</span><span class="n">x</span><span class="o">==</span><span class="n">a</span><span class="o">[</span><span class="n">middle</span><span class="o">])</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">middle</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">if</span><span class="o">(</span><span class="n">x</span><span class="o">></span><span class="n">a</span><span class="o">[</span><span class="n">middle</span><span class="o">])</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">left</span><span class="o">=</span><span class="n">middle</span><span class="o">+</span><span class="mi">1</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">else</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">right</span><span class="o">=</span><span class="n">middle</span><span class="o">-</span><span class="mi">1</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">return</span><span class="o">-</span><span class="mi">1</span><span class="o">;</span><span class="c1">//未找到x
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="o">}</span>
|
||
</span></span></code></pre></td></tr></table></div>
|
||
</div>
|
||
</div></li>
|
||
</ol>
|
||
<h5 id="棋盘覆盖"><a href="#棋盘覆盖" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>棋盘覆盖</h5>
|
||
<p>在一个$2^k\times2^k$ 个方格组成的棋盘中，恰有一个方格与其他方格不同，称该方格为一特殊方格，且称该棋盘为一特殊棋盘,在该问题中要用4中种不同形态的L型骨牌覆盖给定的特殊棋盘上除特殊方格以外的所有方格，且任意2个L型骨牌不得重叠覆盖</p>
|
||
<p>利用分治法得出的简洁算法</p>
|
||
<p>当k>0时，将$ 2^k\times 2^k $棋盘分割为4个$ 2^{k-1}\times 2^{k-1}$ 子棋盘，特殊方格必定位于4个较小子棋盘之一，其余3个子棋盘中无特殊方格。未来将这3个无特殊方格的子棋盘转化为特殊棋盘，可以用一个L型骨牌覆盖着3个较小棋盘的会合处，这3个子棋盘上被L型骨牌覆盖的方格就成为该棋盘上的特殊方格，从而将原问题转化为4个较小规模的棋盘覆盖问题，递归地使用这种分割，直至棋盘简化为1</p>
|
||
<div class="highlight"><div class="chroma">
|
||
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|
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<td class="lntd">
|
||
<pre tabindex="0" class="chroma"><code class="language-java" data-lang="java"><span class="line"><span class="cl"><span class="kd">public</span> <span class="kd">class</span> <span class="nc">Divide_and_conquer</span> <span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kd">public</span> <span class="kt">void</span> <span class="nf">chessBoard</span><span class="o">(</span><span class="kt">int</span> <span class="n">tr</span><span class="o">,</span><span class="kt">int</span> <span class="n">tc</span><span class="o">,</span><span class="kt">int</span> <span class="n">dr</span><span class="o">,</span><span class="kt">int</span> <span class="n">dc</span><span class="o">,</span><span class="kt">int</span> <span class="n">size</span><span class="o">){</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kt">int</span> <span class="n">tile</span><span class="o">=</span><span class="mi">0</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kt">int</span> <span class="n">board</span><span class="o">[][]=</span><span class="k">new</span> <span class="kt">int</span><span class="o">[</span><span class="mi">6</span><span class="o">][</span><span class="mi">6</span><span class="o">];</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">if</span><span class="o">(</span><span class="n">size</span><span class="o">==</span><span class="mi">1</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">return</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kt">int</span> <span class="n">t</span><span class="o">=</span><span class="n">tile</span><span class="o">++,</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">s</span><span class="o">=</span><span class="n">size</span><span class="o">/</span><span class="mi">2</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖左上角棋盘
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="k">if</span><span class="o">(</span><span class="n">dr</span><span class="o"><</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">&&</span><span class="n">dc</span><span class="o"><</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">,</span> <span class="n">tc</span><span class="o">,</span> <span class="n">dr</span><span class="o">,</span> <span class="n">dc</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">else</span><span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//用t号L型骨牌覆盖右下角
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">board</span><span class="o">[</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">][</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">]=</span><span class="n">t</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖其余方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">,</span><span class="n">tc</span><span class="o">,</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">,</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">,</span><span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖右上角子棋盘
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="k">if</span><span class="o">(</span><span class="n">dr</span><span class="o"><</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">&&</span><span class="n">dc</span><span class="o">>=</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//特殊方格在此棋盘中
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">,</span> <span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">dr</span><span class="o">,</span> <span class="n">dc</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">else</span><span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//此棋盘中无特殊方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="c1">//用t号L型骨牌覆盖左下角
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">board</span><span class="o">[</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">][</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">]=</span><span class="n">t</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖其他方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">,</span> <span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">,</span> <span class="n">tc</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖左上角棋盘
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="k">if</span><span class="o">(</span><span class="n">dr</span><span class="o">>=</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">&&</span><span class="n">dc</span><span class="o"><</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//特殊方格在此棋盘中
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tc</span><span class="o">,</span> <span class="n">dr</span><span class="o">,</span> <span class="n">dc</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">else</span><span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//此棋盘中无特殊方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="c1">//用t号L型骨牌覆盖右下角
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">board</span><span class="o">[</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">][</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">]=</span><span class="n">t</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖其他方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tc</span><span class="o">,</span> <span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖右下角子棋盘
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="k">if</span><span class="o">(</span><span class="n">dr</span><span class="o">>=</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">&&</span><span class="n">dc</span><span class="o">>=</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//特殊方格在此棋盘中
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">dr</span><span class="o">,</span> <span class="n">dc</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">else</span><span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//此棋盘中无特殊方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="c1">//用t号L型骨牌覆盖左下角
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">board</span><span class="o">[</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">][</span><span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">]=</span><span class="n">t</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//覆盖其他方格
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="n">chessBoard</span><span class="o">(</span><span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tr</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">tc</span><span class="o">+</span><span class="n">s</span><span class="o">,</span> <span class="n">s</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="o">}</span>
|
||
</span></span></code></pre></td></tr></table></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
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</article>
|
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<div class="updated-badge-container">
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<span title="Updated @ 2023-06-18 21:19:01 CST" style="cursor:help">
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