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<h2 class="post-title" itemprop="name headline">Binomial Heap
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<p><strong>Binomial Heap</strong>是由一群 <strong>Binomail Tree</strong>所組成的<br><strong>Binomial Tree(BT)</strong>含有下列特性</p>
<ul>
<li>高度為k的 BT共有2^k個node</li>
<li>高度為k的 BT可以看成 BT0~BTk-1的組合 再加上一個root組成<br><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Binomial_Trees.svg/700px-Binomial_Trees.svg.png" alt=""></li>
</ul>
<a id="more"></a>
<p><strong>Binomial Heap</strong></p>
<ul>
<li>是 <strong>mergable heap</strong></li>
<li>由一群 <strong>Binomial Tree</strong>組成，每個BT都滿足 min-heap的性質</li>
<li>對於高度為k的BT只能存在最多一棵</li>
<li>以二進位來看待的話，第K位就代表是否存在高度為K的BT<ul>
<li>以下圖為例，就是11001 (右邊最小)<ul>
<li>因此任何數量的結點都可以用不同的BT給組合出來</li>
</ul>
</li>
</ul>
</li>
</ul>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Binomial-heap-13.svg/498px-Binomial-heap-13.svg.png" alt=""></p>
<p>##Implement##</p>
<ul>
<li>採用 <strong>Left-Child Right-sibling</strong>的方式來實現，左邊指向child,右邊指向同輩</li>
<li>value: node的值</li>
<li>degree: 以此node為root的BT的高度</li>
<li>parent: 指向其parent<br><img src="http://user-image.logdown.io/user/415/blog/415/post/173103/3x9u0iDPRN606rAy5Ir7_%E8%9E%A2%E5%B9%95%E6%88%AA%E5%9C%96%202014-01-03%2021.42.57.png" alt="螢幕截圖 2014-01-03 21.42.57.png"><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Node</span>{</span></span><br><span class="line"> <span class="keyword">public</span>:</span><br><span class="line"> Node* parent;</span><br><span class="line"> Node* child;</span><br><span class="line"> Node* sibling;</span><br><span class="line"> <span class="keyword">int</span> value;</span><br><span class="line"> <span class="keyword">int</span> degree;</span><br><span class="line"> Node(){</span><br><span class="line"> parent = <span class="literal">NULL</span>;</span><br><span class="line"> child = <span class="literal">NULL</span>;</span><br><span class="line"> sibling = <span class="literal">NULL</span>;</span><br><span class="line"> value = <span class="number">0</span>;</span><br><span class="line"> degree = <span class="number">0</span>;</span><br><span class="line"> }</span><br><span class="line">};</span><br></pre></td></tr></table></figure>
</li>
</ul>
<p>##Functions##</p>
<ul>
<li>getMin</li>
<li>size</li>
<li>Travese (postorder)</li>
<li>mergeHeap</li>
<li>Insert</li>
<li>deleteMin</li>
</ul>
<p>##getMin##<br>由於每個BT本身都已經是min-heap的特性了，因此只要針對每個BT的root比較其值即可<br><figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getMin</span><span class="params">()</span></span>{</span><br><span class="line"> Node* x = head;</span><br><span class="line"> <span class="keyword">int</span> min = INT_MAX; </span><br><span class="line"> <span class="keyword">while</span>(x!=<span class="literal">NULL</span>){</span><br><span class="line"> <span class="keyword">if</span>(x->value < min)</span><br><span class="line"> min = x->value;</span><br><span class="line"> x = x->sibling;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> min;</span><br><span class="line">}</span><br></pre></td></tr></table></figure></p>
<p>##size##<br>由於 <strong>Binomial Heap</strong>內都是由 <strong>Binomial Tree</strong>組成，所以可以由每個BT的degree得到其node數量<br>再把所有加總即可。</p>
<figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">size</span><span class="params">()</span></span>{</span><br><span class="line"> Node* tmp = head;</span><br><span class="line"> <span class="keyword">int</span> count=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">while</span>(tmp){</span><br><span class="line"> count+= (<span class="number">1</span><<tmp->degree); <span class="comment">// 2^degree</span></span><br><span class="line"> tmp = tmp->sibling;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> count;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
<p>##Postorder##<br>這邊是每個BT都要獨立跑一次Postorder的結果，所以在遞迴的過程中要對root做一些控制<br><figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//對每一棵BT都跑一次postorder</span></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">postorder</span><span class="params">()</span></span>{</span><br><span class="line"> Node* tmp = head;</span><br><span class="line"> <span class="keyword">while</span>(tmp){</span><br><span class="line"> _postorder(tmp);</span><br><span class="line"> tmp = tmp->sibling;</span><br><span class="line">}</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"\n"</span>);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="comment">//用parent判斷是不是root,避免root跑去呼叫到別的BT</span></span><br><span class="line"><span class="keyword">void</span> _postorder(Node* node){</span><br><span class="line"> <span class="keyword">if</span>(!node)</span><br><span class="line"> <span class="keyword">return</span>;</span><br><span class="line"> _postorder(node->child);</span><br><span class="line"> <span class="keyword">if</span>(node->parent)</span><br><span class="line"> _postorder(node->sibling);</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%d "</span>,node->value);</span><br><span class="line">}</span><br><span class="line">``` </span><br><span class="line">##MergeHeap##</span><br><span class="line">要合併兩個 **Binomial Heap**</span><br><span class="line">- 先把兩個 **Binomail Heap**的 BT <span class="built_in">list</span>給重新串接起來，以degree為key做sorting.</span><br><span class="line">- 再根據這個新的BT <span class="built_in">list</span>開始進行一系列的合併</span><br><span class="line">- 如果只有兩個高度相同的BT，就直接合併</span><br><span class="line">- 如果有三個高度相同的BT，就把後面兩棵合併(維持sorting)</span><br><span class="line"></span><br><span class="line">``` c</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">MergeHeap</span><span class="params">(BinomialHeap &bh)</span></span>{</span><br><span class="line"> </span><br><span class="line"> mergeHeap(bh); <span class="comment">//先把BT list給重新串接起來</span></span><br><span class="line"> Node* prev = <span class="literal">NULL</span>;</span><br><span class="line"> Node* x = head;</span><br><span class="line"> Node* next = x->sibling;</span><br><span class="line"> <span class="keyword">while</span>(next){</span><br><span class="line"> <span class="keyword">if</span>( (x->degree != next->degree) || next->sibling && next->sibling->degree == x->degree){</span><br><span class="line"> prev = x; <span class="comment">//前後兩棵BT的高度不同 或是 後面三棵BT的高度都相同</span></span><br><span class="line"> x = next; <span class="comment">//那就把指標往前移動，下次再合併</span></span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">if</span>( x->value <= next->value){ <span class="comment">//前面BT的值比較小，所以後面的合併進來</span></span><br><span class="line"> x->sibling = next->sibling; </span><br><span class="line"> mergeTree(next,x); </span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span>{ <span class="comment">//前面那棵BT的值比較大，要往後合併，視情況也要更新 head指標</span></span><br><span class="line"> <span class="keyword">if</span>(!prev){ </span><br><span class="line"> head = next; <span class="comment">//更新head 指標</span></span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span>{</span><br><span class="line"> prev->sibling = next; </span><br><span class="line"> }</span><br><span class="line"> mergeTree(x,next); <span class="comment">//合併</span></span><br><span class="line"> x = next; </span><br><span class="line"> }</span><br><span class="line"> next = next->sibling; </span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure></p>
<p>要把兩個 <strong>Binomial Heap</strong>的BT list給重新串接起來，採用 <strong>merge sort的方法</strong><br><img src="http://user-image.logdown.io/user/415/blog/415/post/173103/4wzFb15nSdmxfVoiRPEm_%E8%9E%A2%E5%B9%95%E6%88%AA%E5%9C%96%202014-01-03%2022.49.08.png" alt="螢幕截圖 2014-01-03 22.49.08.png"></p>
<ul>
<li>使用 <strong>newHead</strong>紀錄合併後的頭</li>
<li>使用 <strong>newCurr</strong>來紀錄每次合併後的尾</li>
</ul>
<figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">mergeHeap</span><span class="params">(BinomialHeap &bh)</span></span>{</span><br><span class="line"> Node* head2 = bh.head;</span><br><span class="line"> Node* head1 = head;</span><br><span class="line"> </span><br><span class="line"> Node* newHead, *newCurr;</span><br><span class="line"></span><br><span class="line"> <span class="keyword">if</span>(!head1){ <span class="comment">//如果本身是空的，就不需要合併，直接指向對方即可</span></span><br><span class="line"> head = head2;</span><br><span class="line"> <span class="keyword">return</span> ;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">if</span>(!head2){ <span class="comment">//對方是空的，也不需要合併</span></span><br><span class="line"> <span class="keyword">return</span> ;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">//先行尋找誰的開頭比較小，當做新串列的頭</span></span><br><span class="line"> <span class="keyword">if</span>(head1->degree > head2->degree){</span><br><span class="line"> newHead = newCurr = head2;</span><br><span class="line"> head2 = head2->sibling;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span> {</span><br><span class="line"> newHead = newCurr = head1;</span><br><span class="line"> head1 = head1->sibling;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="keyword">while</span>(head1 && head2){</span><br><span class="line"> <span class="keyword">if</span>(head1->degree < head2->degree){</span><br><span class="line"> newCurr->sibling = head1;</span><br><span class="line"> newCurr = head1;</span><br><span class="line"> head1 = head1->sibling;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span> {</span><br><span class="line"> newCurr->sibling = head2;</span><br><span class="line"> newCurr = head2;</span><br><span class="line"> head2 = head2->sibling;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">while</span>(head1){</span><br><span class="line"> newCurr->sibling = head1;</span><br><span class="line"> newCurr = head1;</span><br><span class="line"> head1 = head1->sibling;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">while</span>(head2){</span><br><span class="line"> newCurr->sibling = head2;</span><br><span class="line"> newCurr = head2;</span><br><span class="line"> head2 = head2->sibling;</span><br><span class="line"> }</span><br><span class="line"> </span><br><span class="line"> head = newHead;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
<p>合併兩個 <strong>Binomial Tree</strong>，由於我們是min-heap的特性，所以當兩棵高度相等的BT要合併時，根據root的值來決定誰是合併後的root.</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Binomial_heap_merge1.svg/275px-Binomial_heap_merge1.svg.png" alt=""></p>
<p>假設已經知道BT(y)的值比BT(z)還要大，所以BT(z)會是合併後的root</p>
<ul>
<li>y的parent指到z</li>
<li>y的sibling 指到 z本來的child</li>
<li>z的child 指到y</li>
<li>z的degree 加一</li>
</ul>
<figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">mergeTree</span><span class="params">(Node* y,Node* z)</span></span>{</span><br><span class="line"> y->parent = z;</span><br><span class="line"> y->sibling = z->child;</span><br><span class="line"> z->child = y;</span><br><span class="line"> z->degree++;</span><br><span class="line">}</span><br><span class="line">``` </span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">##Insert##</span><br><span class="line">要插入一個新的元素，就是創見一個新的 **Binomial Heap**，然後跟原本的Heap執行合併即可</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">``` c</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">insert</span><span class="params">(<span class="keyword">int</span> value)</span></span>{</span><br><span class="line"> BinomialHeap bh;</span><br><span class="line"> bh.head = <span class="keyword">new</span> Node();</span><br><span class="line"> bh.head->value = value;</span><br><span class="line"> MergeHeap(bh);</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
<p>##Delete##<br>要從 <strong>BinomialHeap</strong>中刪除當前最小元素</p>
<ul>
<li>先找到最小元素所在的那棵BT</li>
<li>把該BT從list裡面拔除</li>
<li>把該BT的children給反向排序(degree為key)</li>
<li>在跟原本的BT list合併</li>
</ul>
<p><img src="http://user-image.logdown.io/user/415/blog/415/post/173103/IsA2xggSb2hfoKuPJ4N6_%E8%9E%A2%E5%B9%95%E6%88%AA%E5%9C%96%202014-01-03%2023.02.35.png" alt="螢幕截圖 2014-01-03 23.02.35.png"></p>
<figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">deleteMin</span><span class="params">()</span></span>{</span><br><span class="line"> <span class="keyword">int</span> min = head->value;</span><br><span class="line"> Node* tmp = head;</span><br><span class="line"> Node* minPre = <span class="literal">NULL</span>;</span><br><span class="line"> Node* minCurr = head;</span><br><span class="line"> <span class="comment">// 找到最小的node位於何處，由於要將該BT給拔除，所以必須要記得該BT前面那棵BT</span></span><br><span class="line"> <span class="comment">// 如果最小棵的是第一棵，那也要一併更新 head 指標</span></span><br><span class="line"> <span class="keyword">while</span>(tmp->sibling){</span><br><span class="line"> <span class="keyword">if</span>(tmp->sibling->value < min){</span><br><span class="line"> min = tmp->sibling->value;</span><br><span class="line"> minPre = tmp;</span><br><span class="line"> minCurr = tmp->sibling;</span><br><span class="line"> }</span><br><span class="line"> tmp = tmp->sibling;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span>(!minPre && minCurr) <span class="comment">//最小棵是第一個</span></span><br><span class="line"> head = minCurr->sibling;</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">if</span>(minPre && minCurr)</span><br><span class="line"> minPre->sibling = minCurr->sibling;</span><br><span class="line"> </span><br><span class="line"> <span class="comment">//H' Make-BINOMIAL-HEAP()</span></span><br><span class="line"> </span><br><span class="line"> Node *pre,*curr;</span><br><span class="line"> <span class="comment">//用三個指標反轉一個 single link list</span></span><br><span class="line"> pre = tmp = <span class="literal">NULL</span>;</span><br><span class="line"> curr = minCurr->child;</span><br><span class="line"> <span class="keyword">while</span>(curr){</span><br><span class="line"> tmp = curr->sibling;</span><br><span class="line"> curr->sibling = pre;</span><br><span class="line"> curr->parent = <span class="literal">NULL</span>;</span><br><span class="line"> pre = curr;</span><br><span class="line"> curr = tmp;</span><br><span class="line"> }</span><br><span class="line"> <span class="comment">//創建一棵新的binomial heap,並且讓他的head 指向反轉後的BT list</span></span><br><span class="line"> BinomialHeap bh ;</span><br><span class="line"> bh.head = pre;</span><br><span class="line"> <span class="comment">//再度合併</span></span><br><span class="line"> MergeHeap(bh);</span><br><span class="line"></span><br><span class="line">}</span><br></pre></td></tr></table></figure>
<p>圖片來自 </p>
<ol>
<li><a href="http://en.wikipedia.org/wiki/Binomial_heap" target="_blank" rel="noopener">Binomial Wiki</a></li>
<li>Introduction To Algorithms，Chapter 19 Binomial Heaps</li>
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