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<article class="content post h-entry" data-align="justify" data-type="algorithm" data-toc-num="true">
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<h1 class="post-title p-name">递归</h1>
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<div class="post-body e-content">
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<ul>
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<li>递归技术
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<ul>
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<li>直接或间接地调用自身的算法称为递归算法</li>
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<li>用函数自身给出定义的函数称为递归函数</li>
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<li>每个递归函数都必须有非递归定义的初始值，否则递归函数无法计算</li>
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</ul>
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</li>
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</ul>
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<hr>
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<ul>
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<li>阶乘函数
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<ul>
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<li>
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<p>可递归地定义为$n!= \begin{cases}1,& \text{n=0} \\ n(n-1)!,&\text{n>0} \end{cases}$</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></code></pre></td>
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<td class="lntd">
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<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">Factorial</span><span class="o">(</span><span class="kt">int</span> <span class="n">n</span><span class="o">){</span>
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</span></span><span class="line"><span class="cl"> <span class="k">if</span><span class="o">(</span><span class="n">n</span><span class="o">==</span><span class="mi">0</span><span class="o">)</span>
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</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="mi">1</span><span class="o">;</span>
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</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">n</span><span class="o">*</span><span class="n">Factorial</span><span class="o">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="o">);</span>
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</span></span><span class="line"><span class="cl"> <span class="o">}</span>
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</span></span></code></pre></td></tr></table></div>
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</div>
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</div></li>
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</ul>
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</li>
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</ul>
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<hr>
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<ul>
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<li>
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<p>Fibonacci数列</p>
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<ul>
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<li>可递归地定义为$F(n)=\begin{cases}1, &\text{n=0},1 \\ F(n-1)+F(n-2) &\text{n>1} \end{cases}$</li>
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</ul>
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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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<td class="lntd">
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<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">fibonacci</span><span class="o">(</span><span class="kt">int</span> <span class="n">n</span><span class="o">){</span>
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</span></span><span class="line"><span class="cl"> <span class="k">if</span><span class="o">(</span><span class="n">n</span><span class="o"><=</span><span class="mi">1</span><span class="o">)</span>
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</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="mi">1</span><span class="o">;</span>
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</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">fibonacci</span><span class="o">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="o">)+</span><span class="n">fibonacci</span><span class="o">(</span><span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="o">);</span>
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</span></span><span class="line"><span class="cl"> <span class="o">}</span>
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</span></span></code></pre></td></tr></table></div>
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</div>
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</div></li>
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</ul>
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<hr>
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<ul>
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<li>
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<p>排列问题</p>
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<ul>
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<li>
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<p>当$n=1时，perm(R)=(r)$,其中r是集合R中唯一的元素；</p>
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</li>
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<li>
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<p>$当n>1时,perm(R)=(r)$
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$perm(R)由(r_1)perm(R_1),(r_2)perm(R_2),\cdots,(r_n)perm(R_n)构成$</p>
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</li>
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<li>
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<p>算法perm(list,k,m)递归地产生所有前缀是list[0:k-1],且后缀是list[k:m]的全排列的所有排列。调用perm(list,0,n-1)即产生list[0:n-1]的全排列</p>
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</li>
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<li>
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<p>一般情况下，$k<m$</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"> 3
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</span><span class="lnt"> 6
|
||
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|
||
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|
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|
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|
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|
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|
||
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|
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|
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|
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|
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|
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|
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|
||
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|
||
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|
||
</span><span class="lnt">22
|
||
</span><span class="lnt">23
|
||
</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">void</span> <span class="nf">perm</span><span class="o">(</span><span class="n">Object</span><span class="o">[]</span> <span class="n">list</span><span class="o">,</span><span class="kt">int</span> <span class="n">k</span><span class="o">,</span><span class="kt">int</span> <span class="n">m</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">k</span><span class="o">==</span><span class="n">m</span><span class="o">){</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//只剩一个元素
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="k">for</span><span class="o">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="o">;</span><span class="n">i</span><span class="o"><=</span><span class="n">m</span><span class="o">;</span><span class="n">i</span><span class="o">++)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">print</span><span class="o">(</span><span class="n">list</span><span class="o">[</span><span class="n">i</span><span class="o">]);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</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">else</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="c1">//还有多个元素，递归产生排列
|
||
</span></span></span><span class="line"><span class="cl"><span class="c1"></span> <span class="k">for</span><span class="o">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="n">k</span><span class="o">;</span><span class="n">i</span><span class="o"><=</span><span class="n">m</span><span class="o">;</span><span class="n">i</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="n">MyMath</span><span class="o">.</span><span class="na">swap</span><span class="o">(</span><span class="n">list</span><span class="o">,</span><span class="n">k</span><span class="o">,</span><span class="n">i</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">perm</span><span class="o">(</span><span class="n">list</span><span class="o">,</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="o">,</span><span class="n">m</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">MyMath</span><span class="o">.</span><span class="na">swap</span><span class="o">(</span><span class="n">list</span><span class="o">,</span><span class="n">k</span><span class="o">,</span><span class="n">i</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="o">}</span>
|
||
</span></span><span class="line"><span class="cl"><span class="kd">public</span> <span class="kd">static</span> <span class="kd">class</span> <span class="nc">MyMath</span><span class="o">{</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kd">public</span> <span class="kd">static</span> <span class="kt">void</span> <span class="nf">swap</span><span class="o">(</span><span class="n">Object</span><span class="o">[]</span> <span class="n">list</span><span class="o">,</span><span class="kt">int</span> <span class="n">k</span><span class="o">,</span><span class="kt">int</span> <span class="n">m</span><span class="o">){</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="kt">int</span> <span class="n">temp</span> <span class="o">=</span> <span class="n">k</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">k</span> <span class="o">=</span> <span class="n">m</span><span class="o">;</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">m</span> <span class="o">=</span> <span class="n">temp</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="o">}</span>
|
||
</span></span></code></pre></td></tr></table></div>
|
||
</div>
|
||
</div></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<hr>
|
||
<ul>
|
||
<li>
|
||
<p>整数划分问题</p>
|
||
<ul>
|
||
<li>
|
||
<p>将正整数$n$表示成一系列正整数之和，$n=n_1+n_2+...+n_k$,其中$n_1\geq n_2\geq ...\geq n_k\geq 1,k\geq 1$</p>
|
||
</li>
|
||
<li>
|
||
<p>将最大加数$n_1$不大于$m$的划分个数记作$q(n,m)$可建立如下递归关系</p>
|
||
<ul>
|
||
<li>当最大加数$n_1$不大于1时，任何正整数n只有一种划分形式，即$n=\begin{matrix} n \\ \overbrace{1+1+\cdots+1}\end{matrix}$</li>
|
||
<li>最大加数$n_1$实际上不能大于$n$。因此，$q(1,m)=1$。</li>
|
||
<li>正整数$n$的划分由$n_1=n$的划分和$n_1\leq n-1$的划分组成</li>
|
||
<li>正整数$n$的最大加数$n_1$不大于$m$的划分由$n_1=m$的划分和$n_1\leq m-1$的划分组成</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<div class="highlight"><div class="chroma">
|
||
<div class="table-container"><table class="lntable"><tr><td class="lntd">
|
||
<pre tabindex="0" class="chroma"><code><span class="lnt"> 1
|
||
</span><span class="lnt"> 2
|
||
</span><span class="lnt"> 3
|
||
</span><span class="lnt"> 4
|
||
</span><span class="lnt"> 5
|
||
</span><span class="lnt"> 6
|
||
</span><span class="lnt"> 7
|
||
</span><span class="lnt"> 8
|
||
</span><span class="lnt"> 9
|
||
</span><span class="lnt">10
|
||
</span><span class="lnt">11
|
||
</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">q</span><span class="o">(</span><span class="kt">int</span> <span class="n">n</span><span class="o">,</span><span class="kt">int</span> <span class="n">m</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">n</span><span class="o"><</span><span class="mi">1</span><span class="o">)||(</span><span class="n">m</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="mi">0</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">n</span><span class="o">==</span><span class="mi">1</span><span class="o">)||(</span><span class="n">m</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="mi">1</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">n</span><span class="o"><</span><span class="n">m</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">q</span><span class="o">(</span><span class="n">n</span><span class="o">,</span><span class="n">n</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">n</span><span class="o">==</span><span class="n">m</span><span class="o">)</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">q</span><span class="o">(</span><span class="n">n</span><span class="o">,</span><span class="n">m</span><span class="o">-</span><span class="mi">1</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="n">q</span><span class="o">(</span><span class="n">n</span><span class="o">,</span><span class="n">m</span><span class="o">-</span><span class="mi">1</span><span class="o">)+</span><span class="n">q</span><span class="o">(</span><span class="n">n</span><span class="o">-</span><span class="n">m</span><span class="o">,</span><span class="n">m</span><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></li>
|
||
</ul>
|
||
<hr>
|
||
<ul>
|
||
<li>
|
||
<p>Hanoi塔问题</p>
|
||
<ul>
|
||
<li>
|
||
<p>设a,b,c是三个塔座。开始是在塔座a上有一叠共n个圆盘，这些圆盘自下而上，由大到小地叠在一起。各圆盘从小到大编号为1，2，···，n。现要求将塔座a上的这一叠圆盘移到塔座b上，并仍按同样顺序叠置。在移动圆盘时应该遵守以下移动规则。</p>
|
||
<ul>
|
||
<li>每次只移动一个圆盘</li>
|
||
<li>任何时刻都不允许将较大的圆盘压在较小的圆盘之上。</li>
|
||
<li>在满足前两个规则的前提下，可将圆盘移至a，b，c任一塔座上</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>递归关系</p>
|
||
<ul>
|
||
<li>$n=1$时，将编号为一的圆盘从塔座a直接移至塔座b上即可。</li>
|
||
<li>$n>1$时，需要利用塔座c作为辅助塔座，
|
||
<ul>
|
||
<li>将$n-1$个较小的圆盘依照移动规则从塔座a移至塔座c，</li>
|
||
<li>将剩下的最大圆盘从塔座a移至塔座b，</li>
|
||
<li>将$n-1$个较小的圆盘依照移动规则从塔座c移至塔座b</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<div class="highlight"><div class="chroma">
|
||
<div class="table-container"><table class="lntable"><tr><td class="lntd">
|
||
<pre tabindex="0" class="chroma"><code><span class="lnt"> 1
|
||
</span><span class="lnt"> 2
|
||
</span><span class="lnt"> 3
|
||
</span><span class="lnt"> 4
|
||
</span><span class="lnt"> 5
|
||
</span><span class="lnt"> 6
|
||
</span><span class="lnt"> 7
|
||
</span><span class="lnt"> 8
|
||
</span><span class="lnt"> 9
|
||
</span><span class="lnt">10
|
||
</span><span class="lnt">11
|
||
</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">void</span> <span class="nf">hanoi</span><span class="o">(</span><span class="kt">int</span> <span class="n">n</span><span class="o">,</span><span class="kt">char</span> <span class="n">a</span><span class="o">,</span><span class="kt">char</span> <span class="n">b</span><span class="o">,</span><span class="kt">char</span> <span class="n">c</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">n</span><span class="o">></span><span class="mi">0</span><span class="o">){</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">hanoi</span><span class="o">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="o">,</span> <span class="n">a</span><span class="o">,</span> <span class="n">c</span><span class="o">,</span> <span class="n">b</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">move</span><span class="o">(</span><span class="n">n</span><span class="o">,</span><span class="n">a</span><span class="o">,</span><span class="n">b</span><span class="o">);</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">hanoi</span><span class="o">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="o">,</span><span class="n">c</span><span class="o">,</span><span class="n">b</span><span class="o">,</span><span class="n">a</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="kd">public</span> <span class="kd">static</span> <span class="kt">void</span> <span class="nf">move</span><span class="o">(</span><span class="kt">int</span> <span class="n">n</span><span class="o">,</span><span class="kt">char</span> <span class="n">a</span><span class="o">,</span><span class="kt">char</span> <span class="n">b</span><span class="o">){</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="s">"第"</span><span class="o">+</span><span class="n">n</span><span class="o">+</span><span class="s">"个盘子从"</span><span class="o">+</span><span class="n">a</span><span class="o">+</span><span class="s">"--->"</span><span class="o">+</span><span class="n">b</span><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></li>
|
||
<li>
|
||
<p>递归调用总结和系统原理</p>
|
||
<ul>
|
||
<li>实现递归调用的关键：为算法建立递归调用工作栈</li>
|
||
<li>运行被调用算法前的行为
|
||
<ul>
|
||
<li>为所有实参指针，返回地址等信息传递给被调用算法</li>
|
||
<li>为被调用算法的局部变量分配存储区</li>
|
||
<li>将控制转移到被调用算法的入口</li>
|
||
</ul>
|
||
</li>
|
||
<li>从被调用算法返回调用算法时
|
||
<ul>
|
||
<li>保存被调用算法的计算结果</li>
|
||
<li>释放分配给被调用算法的数据区</li>
|
||
<li>依照被调用算法保存的返回地址将控制转移到调用算法</li>
|
||
</ul>
|
||
</li>
|
||
<li>嵌套调用时的系统原则：后调用先返回，即算法间的信息传递和控制转移通过栈来实现</li>
|
||
<li>递归算法的调用层次
|
||
<ul>
|
||
<li>调用一个递归算法的主算法为第0层算法</li>
|
||
<li>从主算法调用递归算法为进入第1层调用</li>
|
||
<li>从第i层递归调用本算法为进入第i+1层调用。</li>
|
||
<li>退出第i层递归调用，则返回至i-1层调用。</li>
|
||
</ul>
|
||
</li>
|
||
<li>递归调用的栈使用情况示意</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<div class="table-container"><table>
|
||
<thead>
|
||
<tr>
|
||
<th style="text-align:center">主算法栈块</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:center">M</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center">主算法调用递归算法A的栈块</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center">算法A的第一层递归调用工作记录</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center">算法A的第二层递归调用工作记录</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center">TOP</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center">M</td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
|
||
</article>
|
||
|
||
|
||
|
||
|
||
<div class="updated-badge-container">
|
||
<span title="Updated @ 2023-06-18 21:19:01 CST" style="cursor:help">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" width="130" height="20" class="updated-badge"><linearGradient id="b" x2="0" y2="100%"><stop offset="0" stop-color="#bbb" stop-opacity=".1"/><stop offset="1" stop-opacity=".1"/></linearGradient><clipPath id="a"><rect width="130" height="20" rx="3" fill="#fff"/></clipPath><g clip-path="url(#a)"><path class="updated-badge-left" d="M0 0h55v20H0z"/><path class="updated-badge-right" d="M55 0h75v20H55z"/><path fill="url(#b)" d="M0 0h130v20H0z"/></g><g fill="#fff" text-anchor="middle" font-size="110"><text x="285" y="150" fill="#010101" fill-opacity=".3" textLength="450" transform="scale(.1)">updated</text><text x="285" y="140" textLength="450" transform="scale(.1)">updated</text><text x="915" y="150" fill="#010101" fill-opacity=".3" textLength="650" transform="scale(.1)">2023-06-18</text><text x="915" y="140" textLength="650" transform="scale(.1)">2023-06-18</text></g></svg>
|
||
</span></div>
|
||
|
||
|
||
|
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