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<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<title>Tip</title>
<style>
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<h1 id="tip">Tip</h1>
<p>时间复杂度不仅仅依赖于问题的规模</p>
<p>若算法所需的辅助空间相对输入数据量是个常数，则称算法为原地工作</p>
<h2 id="线性表">线性表</h2>
<ol>
<li>取值插入删除时注意位置的合法性</li>
<li>插入时注意<code>listsize</code>的溢出</li>
<li>双向链表插入删除的时候有4个指针要操作</li>
</ol>
<h2 id="栈">栈</h2>
<ol>
<li><code>Push()</code>时注意栈溢出</li>
<li><code>S.top</code>指向即将push进入的位置</li>
<li><code>realloc()</code>可能会将分配新的空间而非拓展，<strong>返回的指针值可能会变</strong></li>
</ol>
<h2 id="队列">队列</h2>
<ol>
<li>循环队列中<code>Q.rear</code>指向下一个将入队的位置</li>
<li>两种方式处理循环队列
<ol>
<li>设标志判断队空队满</li>
<li>空一个位置，此时分配的空间要比最大队长多1，即如果数组为A[m]，则队列的最长长度为m-1</li>
</ol>
</li>
<li><strong>链队列有头结点</strong>，即front指向头结点而非队头</li>
</ol>
<h2 id="串">串</h2>
<p>定长顺序存储</p>
<ol>
<li>串连接需要考虑截断</li>
<li>求子串考虑起始位置和长度的合法性</li>
</ol>
<p>堆分配存储表示</p>
<ol>
<li>注意要覆盖原串，需要将原串给free</li>
</ol>
<p>模式匹配</p>
<p>KMP</p>
<h2 id="数组">数组</h2>
<p>随机存储结构</p>
<p><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mi>O</mi><mi>C</mi><mo stretchy="false">(</mo><msub><mi>j</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>j</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>j</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi><mi>O</mi><mi>C</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>L</mi><msubsup><mo>∑</mo><mn>1</mn><mi>n</mi></msubsup><msub><mi>j</mi><mi>n</mi></msub><msub><mi>c</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">LOC(j_1, j_2, ...,j_n)=LOC(0,0,...,0)+L\sum\limits_1^{n}j_nc_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">L</span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</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:-0.05724em;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 class="mord mathdefault" style="margin-right:0.05724em;">j</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:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span 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" style="margin-right:0.05724em;">j</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:-0.05724em;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="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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">L</span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</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">0</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:2.3185100000000003em;vertical-align:-0.9671129999999999em;"></span><span class="mord mathdefault">L</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3513970000000004em;"><span style="top:-2.132887em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.0000050000000003em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop op-symbol small-op">∑</span></span></span><span style="top:-3.950005em;margin-left:0em;"><span class="pstrut" style="height:3em;"></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:0.9671129999999999em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</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:-0.05724em;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="mord"><span class="mord mathdefault">c</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></span></span></eq></p>
<p><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>=</mo><msub><mi>c</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>c</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c_n = c_{n-1}b_{n-1}, c_n = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">c</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="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:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><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 class="mbin mtight">−</span><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:0.208331em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><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 class="mbin mtight">−</span><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:0.208331em;"><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">c</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="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></eq></p>
<h2 id="矩阵">矩阵</h2>
<p>稀疏矩阵稀疏因子<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>=</mo><mfrac><mi>t</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></mfrac><mo separator="true">,</mo><mi>t</mi><mtext>为非零元，</mtext><mi>m</mi><mi>n</mi><mtext>为矩阵大小</mtext></mrow><annotation encoding="application/x-tex">\delta = \frac{t}{m\times n}, t为非零元，mn为矩阵大小</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</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:1.227887em;vertical-align:-0.403331em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824556em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mbin mtight">×</span><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">非</span><span class="mord cjk_fallback">零</span><span class="mord cjk_fallback">元</span><span class="mord cjk_fallback">，</span><span class="mord mathdefault">m</span><span class="mord mathdefault">n</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">大</span><span class="mord cjk_fallback">小</span></span></span></span></eq></p>
<p>稀疏矩阵表示方法</p>
<ol>
<li>三元组{i, j, e}</li>
</ol>
<p>三元组的矩阵转置</p>
<ol>
<li>
<p>按顺序查找的方式将在j列的元素放置新表中，然后j += 1</p>
</li>
<li>
<p>对每列第一个元素的恰当位置进行计算，然后以行序为主序遍历原表，放入相应的位置并更新每一列的恰当位置</p>
</li>
<li>
<p>行逻辑链接的顺序表，即加入每行第一个非零元的位置表</p>
</li>
</ol>
<p>矩阵相乘MN</p>
<p>M的每行非零元(i,k)乘N的第k行j列元素加至Q的(i, j)位置，全部加完之后压缩存储</p>
<p>时间复杂度<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi>u</mi><mo>∗</mo><mi>N</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi>u</mi><mi mathvariant="normal">/</mi><mi>N</mi><mi mathvariant="normal">.</mi><mi>m</mi><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(M.tu*N.tu/N.mu)</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="mopen">(</span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord">.</span><span class="mord mathdefault">t</span><span class="mord mathdefault">u</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" style="margin-right:0.10903em;">N</span><span class="mord">.</span><span class="mord mathdefault">t</span><span class="mord mathdefault">u</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mord">.</span><span class="mord mathdefault">m</span><span class="mord mathdefault">u</span><span class="mclose">)</span></span></span></span></eq></p>
<ol start="3">
<li>十字链表right,down,rhead,chead</li>
</ol>
<p>十字链表的初始化与相加</p>
<h2 id="广义表">广义表</h2>
<p>广义表表长为元素个数，表看成一个整体，一个元素</p>
<p>头尾链表空表为null，拓展线性链表还有一个表节点，只是hp，tp都是null</p>
<pre><code><code><div>原广义表:[2, 3, [4, 5], 6]
表头:2 头表:[2] 表尾:[3, [4, 5], 6]
</div></code></code></pre>
<h2 id="树">树</h2>
<h3 id="二叉树">二叉树</h3>
<p>二叉树不是度不大于2的有序树</p>
<p>度不大于2的有序树没有左右孩子之分</p>
<p><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_0=n_2+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">n</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">0</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">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">n</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">2</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:0.2222222222222222em;"></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></eq></p>
<p>n个节点的完全二叉树深度为<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>2</mn></msub><mi>n</mi><mo stretchy="false">]</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">[log_2 n]+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="mopen">[</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</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:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span 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="mord mathdefault">n</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></eq></p>
<p><strong>完全二叉树中度为1的节点最多只有一个</strong></p>
<p>顺序存储编号i的节点存在i-1的位置</p>
<p>链式存储结构</p>
<p>n个节点有n+1个空链域</p>
<p>先序+中序 , 后序+中序 , 层次+中序 可复原二叉树</p>
<p><strong>后序遍历时</strong>，<strong>T的左孩子的直接后继为</strong></p>
<ol>
<li>如果T->rchild <strong>==</strong> NULL,T->lchild->next = T;</li>
<li>如果T->rchild <strong>!=</strong> NULL, 见下代码</li>
</ol>
<pre><code class="language-c"><div>p = T->rchild;
<span class="hljs-keyword">while</span> (p->lchild || p->rchild)
{
<span class="hljs-keyword">if</span> (p->lchild) p = p->lchild;
<span class="hljs-keyword">else</span> p = p->rchild;
}
</div></code></pre>
<p>存储结构</p>
<ol>
<li>双亲表示法，结构体包含双亲</li>
<li>孩子表示法，包含孩子链表的头指针</li>
<li>孩子兄弟表示法</li>
</ol>
<h3 id="霍夫曼树">霍夫曼树</h3>
<p>n个叶子的霍夫曼树有2n-1个节点</p>
<p>通过回溯和试探进行图的遍历</p>
<p>含n个节点的不相似的二叉树共<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><msubsup><mi>C</mi><mrow><mn>2</mn><mi>n</mi></mrow><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\frac{1}{n+1} C_{2n}^{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></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:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4518920000000004em;margin-left:-0.07153em;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 class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span></eq>棵</p>
<h2 id="图">图</h2>
<h3 id="基本概念">基本概念</h3>
<p>网，带权图</p>
<p>a--->b,a弧尾，b弧头</p>
<p>路径P(a,b)</p>
<p>回路P(a, a)</p>
<p>简单路径即图论中的圈</p>
<p>连通分量即图论中的连通片</p>
<p>一个有向图的生成森林由若干棵有向树组成</p>
<h3 id="存储结构">存储结构</h3>
<table>
<thead>
<tr>
<th style="text-align:center">存储结构</th>
<th style="text-align:center">创建时间复杂度</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">邻接矩阵</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2+en)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></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">e</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></eq></td>
</tr>
<tr>
<td style="text-align:center">邻接表</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</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="mopen">(</span><span class="mord mathdefault">n</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">e</span><span class="mclose">)</span></span></span></span></eq>,若输入为顶点编号；<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(ne)</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="mopen">(</span><span class="mord mathdefault">n</span><span class="mord mathdefault">e</span><span class="mclose">)</span></span></span></span></eq>,若输入为顶点，需要查找</td>
</tr>
<tr>
<td style="text-align:center">十字链表</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</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="mopen">(</span><span class="mord mathdefault">n</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">e</span><span class="mclose">)</span></span></span></span></eq>,若输入为顶点编号；<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(ne)</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="mopen">(</span><span class="mord mathdefault">n</span><span class="mord mathdefault">e</span><span class="mclose">)</span></span></span></span></eq>,若输入为顶点，需要查找</td>
</tr>
<tr>
<td style="text-align:center">邻接多重表</td>
<td style="text-align:center">?</td>
</tr>
</tbody>
</table>
<h3 id="图遍历">图遍历</h3>
<p>DFS用邻接矩阵遍历，查找每个顶点的邻接点所需的总时间是<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>∗</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n*n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</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">n</span><span class="mclose">)</span></span></span></span></eq></p>
<p>DFS用邻接表遍历，查找邻接点所需时间为<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e)</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="mopen">(</span><span class="mord mathdefault">e</span><span class="mclose">)</span></span></span></span></eq>，查找顶点需要<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></eq>，总时间<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</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="mopen">(</span><span class="mord mathdefault">n</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">e</span><span class="mclose">)</span></span></span></span></eq></p>
<h3 id="连通性">连通性</h3>
<h4 id="最小生成树">最小生成树</h4>
<p>任何一个无向连通图的最小生成树有一棵或多棵</p>
<p>Prim</p>
<p>找<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo separator="true">,</mo><mover accent="true"><mi>U</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U, \bar{U})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07011em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201099999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">U</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></eq>中代价最小的边，让边代价为0，并且将边的另一个端点加入U中，对这个点的邻边更新代价</p>
<p>时间复杂度<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></eq>，适合边稠密的图</p>
<p>Kruskal</p>
<p>初始条件是n个孤立顶点</p>
<p>选择两个顶点落在不同连通分量上的代价最小的边</p>
<p>时间复杂度<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>l</mi><mi>o</mi><mi>g</mi><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(eloge)</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="mopen">(</span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mclose">)</span></span></span></span></eq>，适合边稀疏</p>
<h4 id="关节点和重连通风量">关节点和重连通风量</h4>
<p>连通度k，至少删去k个顶点才能破坏图的连通性</p>
<p>关节点：</p>
<ol>
<li>有两棵子树的根</li>
<li>非叶子节点v点的子树中所有节点都没有指向v的祖先的边，v为关节点</li>
</ol>
<p><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">{</mo><mi>v</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>v</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>v</mi><mo>−</mo><mo>></mo><mi>p</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo separator="true">,</mo><mi>v</mi><mo>−</mo><mo>></mo><mi>p</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo>−</mo><mo>></mo><mi>p</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>l</mi><mi>o</mi><mi>w</mi><mo stretchy="false">(</mo><mi>v</mi><mo>−</mo><mo>></mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>d</mi><mo separator="true">,</mo><mi>v</mi><mo>−</mo><mo>></mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>d</mi><mo>−</mo><mo>></mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>d</mi><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">low(v) = min\{visited(v), visited(v->parent, v->parent->parent, ...), low(v->child, v->child->child, ...)\}</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:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">m</span><span class="mord mathdefault">i</span><span class="mord mathdefault">n</span><span class="mopen">{</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">i</span><span class="mord mathdefault">s</span><span class="mord mathdefault">i</span><span class="mord mathdefault">t</span><span class="mord mathdefault">e</span><span class="mord mathdefault">d</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">i</span><span class="mord mathdefault">s</span><span class="mord mathdefault">i</span><span class="mord mathdefault">t</span><span class="mord mathdefault">e</span><span class="mord mathdefault">d</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:0.80952em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">e</span><span class="mord mathdefault">n</span><span class="mord mathdefault">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:0.80952em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">e</span><span class="mord mathdefault">n</span><span class="mord mathdefault">t</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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">e</span><span class="mord mathdefault">n</span><span class="mord mathdefault">t</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="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</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="mclose">)</span><span class="mclose">}</span></span></span></span></eq></p>
<p>如果<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mo stretchy="false">(</mo><mi>v</mi><mo>−</mo><mo>></mo><mi>c</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>d</mi><mo stretchy="false">)</mo><mo>></mo><mo>=</mo><mi>v</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">low(v->child) >= visited(v)</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:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></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" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">i</span><span class="mord mathdefault">s</span><span class="mord mathdefault">i</span><span class="mord mathdefault">t</span><span class="mord mathdefault">e</span><span class="mord mathdefault">d</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></eq>，v为割点，如果<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi><mo><</mo><mi>G</mi><mi mathvariant="normal">.</mi><mi>v</mi><mi>e</mi><mi>x</mi><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">count < G.vexnum</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65418em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">u</span><span class="mord mathdefault">n</span><span class="mord mathdefault">t</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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">G</span><span class="mord">.</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">e</span><span class="mord mathdefault">x</span><span class="mord mathdefault">n</span><span class="mord mathdefault">u</span><span class="mord mathdefault">m</span></span></span></span></eq>，根为割点</p>
<h3 id="dag-aov-n-aoe">DAG, AOV-N, AOE</h3>
<p>DAG:有向无环图</p>
<p>AOV-N:顶点活动网络，表示顶点活动的先后顺序</p>
<p>AOE-N:边活动网络，顶点为事件，表示边活动的开始或者结束</p>
<p>将偏序集变为全序即拓扑排序</p>
<p>拓扑排序可检测AOV-N是否有环，通过不断删去入度为0的顶点和更新其他顶点的入度，如果没有剩余，则没有环，否则有环</p>
<p>时间复杂度<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mo>+</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>∗</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e+n+2*n+e)</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="mopen">(</span><span class="mord mathdefault">e</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:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</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">2</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:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</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">e</span><span class="mclose">)</span></span></span></span></eq></p>
<p>关键路径</p>
<p>先按照拓扑排序的方式计算各顶点的最早发生时间，如果拓扑排序失败，说明有环，并将拓扑排序的序列压入栈中，在对逆拓扑排序序列求最迟发生时间</p>
<p>计算弧活动的最早开始时间和最晚开始时间为<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e)</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="mopen">(</span><span class="mord mathdefault">e</span><span class="mclose">)</span></span></span></span></eq>，计算顶点事件的最早和最晚发生时间为<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></eq>，总时间复杂度为<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</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="mopen">(</span><span class="mord mathdefault">n</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">e</span><span class="mclose">)</span></span></span></span></eq></p>
<h3 id="最短路径">最短路径</h3>
<p>Dijkstra算法(一点到其他点的最短路径)：<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo>=</mo><mo stretchy="false">{</mo><msub><mi>v</mi><mn>0</mn></msub><mo stretchy="false">}</mo><mo separator="true">,</mo><mtext>取</mtext><mo stretchy="false">(</mo><mi>S</mi><mo separator="true">,</mo><mover accent="true"><mi>S</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mtext>最短的路径的另一个顶点加入</mtext><mi>S</mi></mrow><annotation encoding="application/x-tex">S_0 = \{ v_0 \}, 取(S, \bar{S})最短的路径的另一个顶点加入S</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" style="margin-right:0.05764em;">S</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:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span 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">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.07011em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</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:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span 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="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">取</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201099999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">短</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">路</span><span class="mord cjk_fallback">径</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">另</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">顶</span><span class="mord cjk_fallback">点</span><span class="mord cjk_fallback">加</span><span class="mord cjk_fallback">入</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span></eq></p>
<p>时间复杂度<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></eq></p>
<p>Floyd(每一对顶点之间的最短路径):找到中间顶点序号<=u的路径，并将其与之前<=u-1的最短路径相比较，然后赋值，最后中间顶点序号<=n-1就是ij之间最短路径</p>
<h2 id="查找">查找</h2>
<p>静态查找表：查询，检索</p>
<p>动态查找表：查询，检索，插入，删除</p>
<p>唯一标识某记录的关键字叫主关键字</p>
<p>平均查找长度ASL：需要和给定关键字进行比较的关键字个数的期望</p>
<table>
<thead>
<tr>
<th style="text-align:center">查找方式</th>
<th style="text-align:center">ASL</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">顺序查找</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{n+1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><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:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></eq></td>
</tr>
<tr>
<td style="text-align:center">折半查找(有序表)</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>n</mi></mfrac><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\frac{n+1}{n} \log_2 n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><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:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</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></eq></td>
</tr>
<tr>
<td style="text-align:center">次优查找树</td>
<td style="text-align:center">构造时间<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></eq></td>
</tr>
<tr>
<td style="text-align:center">索引查找表</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mi>b</mi></msub><mo>+</mo><msub><mi>L</mi><mi>w</mi></msub><mtext>，</mtext><msub><mi>L</mi><mi>b</mi></msub><mtext>为确定所在块的</mtext><mi>A</mi><mi>S</mi><mi>L</mi><mtext>，</mtext><msub><mi>L</mi><mi>w</mi></msub><mtext>为在块中查找的</mtext><mi>A</mi><mi>S</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">L_b+L_w，L_b为确定所在块的ASL，L_w为在块中查找的ASL</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:0.33610799999999996em;"><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">b</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;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 mathdefault mtight" style="margin-right:0.02691em;">w</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="mord cjk_fallback">，</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:0.33610799999999996em;"><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">b</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="mord cjk_fallback">为</span><span class="mord cjk_fallback">确</span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">所</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">块</span><span class="mord cjk_fallback">的</span><span class="mord mathdefault">A</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mord mathdefault">L</span><span class="mord cjk_fallback">，</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" style="margin-right:0.02691em;">w</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="mord cjk_fallback">为</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">块</span><span class="mord cjk_fallback">中</span><span class="mord cjk_fallback">查</span><span class="mord cjk_fallback">找</span><span class="mord cjk_fallback">的</span><span class="mord mathdefault">A</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mord mathdefault">L</span></span></span></span></eq>，顺序查找<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>b</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{b+1}{2}+\frac{s+1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251079999999999em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">+</span><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:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></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:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mbin mtight">+</span><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:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></eq>，折半查找块<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo></mo><mo stretchy="false">(</mo><mfrac><mi>n</mi><mi>s</mi></mfrac><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mfrac><mi>s</mi><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\log(\frac{n}{s}+1)+\frac{s}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></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:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></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">1</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:1.040392em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></eq></td>
</tr>
<tr>
<td style="text-align:center">二叉排序树</td>
<td style="text-align:center">平均性能<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mtext>）</mtext><mi>ln</mi><mo></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\leq2(1+\frac{1}{n}）\ln n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></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">2</span><span class="mopen">(</span><span class="mord">1</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:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></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:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord cjk_fallback">）</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span></span></span></span></eq></td>
</tr>
<tr>
<td style="text-align:center">平衡二叉树</td>
<td style="text-align:center"><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span></span></span></span></eq></td>
</tr>
</tbody>
</table>
<p>斐波那契查找：以斐波那契数列进行分割</p>
<p>插值查找：按照比例查找</p>
<p>次优查找树：取使<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi><mo>=</mo><mi mathvariant="normal">∣</mi><msub><mi mathvariant="normal">Σ</mi><mrow><mi>i</mi><mtext>左边</mtext></mrow></msub><mi>w</mi><mo>−</mo><msub><mi mathvariant="normal">Σ</mi><mrow><mi>i</mi><mtext>右边</mtext></mrow></msub><mi>w</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>s</mi><msub><mi>w</mi><mi>h</mi></msub><mo>+</mo><mi>s</mi><msub><mi>w</mi><mrow><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><mi>s</mi><msub><mi>w</mi><mi>i</mi></msub><mo>−</mo><mi>s</mi><msub><mi>w</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\Delta P = |\Sigma_{i左边} w - \Sigma_{i右边} w| = |sw_{h} + sw_{l-1} - sw_i - sw_{i-1}|</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">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em;">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">∣</span><span class="mord"><span class="mord">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><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">i</span><span class="mord cjk_fallback mtight">左</span><span class="mord cjk_fallback mtight">边</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="mord mathdefault" style="margin-right:0.02691em;">w</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"><span class="mord">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><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">i</span><span class="mord cjk_fallback mtight">右</span><span class="mord cjk_fallback mtight">边</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="mord mathdefault" style="margin-right:0.02691em;">w</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:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;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">h</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.791661em;vertical-align:-0.208331em;"></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;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" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><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:0.208331em;"><span></span></span></span></span></span></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:0.73333em;vertical-align:-0.15em;"></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</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:0.2222222222222222em;"></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">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;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">i</span><span class="mbin mtight">−</span><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:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span></eq>最小的i,然后分治</p>
<p>索引顺序表(分块查找)</p>
<h3 id="二叉排序树二叉查找树">二叉排序树(二叉查找树)</h3>
<h4 id="插入">插入</h4>
<p>在搜索的时候将父亲也给传进去，这样当查找不到(p == null)时可以让p指向父亲，方便插入</p>
<p>在搜索的时候传入父亲null，如果该树为空树，则p=null，需要让T直接指向新创建的节点</p>
<h4 id="删除">删除</h4>
<ol>
<li>右子树为空，直接让左子树的根接替该点</li>
<li>左子树为空，直接让右子树的根接替该点</li>
<li>若左右子树都不空，则让左子树的最右边那个点(该点的直接前驱)代替该点，并且删除左子树最右边的那个点，让那个点的左子树的根代替那个点，如果那个点就是该点左子树的根，则让该点的左孩子为那个点的左子树</li>
</ol>
<h3 id="平衡二叉树avl">平衡二叉树(AVL)</h3>
<p>平衡因子<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>F</mi><mo>=</mo><mi>D</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><mo>></mo><mi>l</mi><mi>c</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>d</mi><mo stretchy="false">)</mo><mo>−</mo><mi>D</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><mo>></mo><mi>r</mi><mi>c</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">BF=Depth(T->lchild) - Depth(T->rchild)=-1,0,1</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" style="margin-right:0.05017em;">B</span><span class="mord mathdefault" style="margin-right:0.13889em;">F</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" style="margin-right:0.02778em;">D</span><span class="mord mathdefault">e</span><span class="mord mathdefault">p</span><span class="mord mathdefault">t</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</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" style="margin-right:0.02778em;">D</span><span class="mord mathdefault">e</span><span class="mord mathdefault">p</span><span class="mord mathdefault">t</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span></span></span></span></eq></p>
<table>
<thead>
<tr>
<th style="text-align:center">类型</th>
<th style="text-align:center">判定</th>
<th style="text-align:center">操作</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">LL</td>
<td style="text-align:center">插入左子树，左子树长高，根bf=1，左孩子bf=1</td>
<td style="text-align:center">单次右旋</td>
</tr>
<tr>
<td style="text-align:center">LR</td>
<td style="text-align:center">插入左子树，左子树长高，根bf=1，左孩子bf=-1</td>
<td style="text-align:center">左旋右旋</td>
</tr>
<tr>
<td style="text-align:center">RR</td>
<td style="text-align:center">插入右子树，右子树长高，根bf=-1，右孩子bf=-1</td>
<td style="text-align:center">单次左旋</td>
</tr>
<tr>
<td style="text-align:center">RL</td>
<td style="text-align:center">插入右子树，右子树长高，根bf=-1，右孩子bf=1</td>
<td style="text-align:center">右旋左旋</td>
</tr>
</tbody>
</table>
<h3 id="b树">B树</h3>
<p>m阶树</p>
<ol>
<li>每个节点最多m棵子树</li>
<li>根节点至少两棵子树</li>
<li>除根外，所有非终端节点至少<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">\lceil\frac{m}{2}\rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mopen">⌈</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌉</span></span></span></span></eq>棵子树</li>
<li>非终端节点<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><msub><mi>A</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>K</mi><mn>1</mn></msub><mo separator="true">,</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>K</mi><mi>n</mi></msub><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n, A_0, K_1, A_1, ..., K_n, A_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">n</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:0.30110799999999993em;"><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">0</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" style="margin-right:0.07153em;">K</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:-0.07153em;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 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" style="margin-right:0.07153em;">K</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:-0.07153em;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"><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="mclose">)</span></span></span></span></eq></li>
<li>所有叶子节点都在同一层，不带信息</li>
</ol>
<h3 id="b树-1">B+树</h3>
<p>m阶B+树</p>
<ol>
<li>n棵子树的节点含n个关键字</li>
<li>所有叶子节点包含所有关键字信息</li>
</ol>
<h3 id="键树数字查找树">键树（数字查找树）</h3>
<h3 id="哈希">哈希</h3>
<p>构造哈希函数</p>
<ol>
<li>直接定址法</li>
<li>除留余数法,最好去素数</li>
</ol>
<p>产生冲突的两个关键字称为<strong>同义词</strong></p>
<p>处理冲突</p>
<ol>
<li><eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mi>k</mi><mi>e</mi><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>d</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>m</mi><mi>o</mi><mi>d</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">H_i = (H(key)+d_i) mod m</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" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</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">=</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="mopen">(</span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</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"><span class="mord mathdefault">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><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">i</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="mclose">)</span><span class="mord mathdefault">m</span><span class="mord mathdefault">o</span><span class="mord mathdefault">d</span><span class="mord mathdefault">m</span></span></span></span></eq>，<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub><mtext>线性探测再散列</mtext><mo>:</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo separator="true">;</mo><mtext>二次探测再散列</mtext><mo>:</mo><msup><mn>1</mn><mn>2</mn></msup><mo separator="true">,</mo><mo>−</mo><msup><mn>1</mn><mn>2</mn></msup><mo separator="true">,</mo><msup><mn>2</mn><mn>2</mn></msup><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msup><mi>k</mi><mn>2</mn></msup><mo separator="true">,</mo><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>≤</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo separator="true">;</mo><mtext>伪随机数列</mtext></mrow><annotation encoding="application/x-tex">d_i线性探测再散列:1, 2, 3, ..., m-1;二次探测再散列:1^2, -1^2, 2^2, ..., k^2, -k^2, (k \leq \frac{m}{2});伪随机数列</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><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">i</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="mord cjk_fallback">线</span><span class="mord cjk_fallback">性</span><span class="mord cjk_fallback">探</span><span class="mord cjk_fallback">测</span><span class="mord cjk_fallback">再</span><span class="mord cjk_fallback">散</span><span class="mord cjk_fallback">列</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</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 mathdefault">m</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">二</span><span class="mord cjk_fallback">次</span><span class="mord cjk_fallback">探</span><span class="mord cjk_fallback">测</span><span class="mord cjk_fallback">再</span><span class="mord cjk_fallback">散</span><span class="mord cjk_fallback">列</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:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="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" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</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:1.095em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">伪</span><span class="mord cjk_fallback">随</span><span class="mord cjk_fallback">机</span><span class="mord cjk_fallback">数</span><span class="mord cjk_fallback">列</span></span></span></span></eq></li>
<li>再哈希</li>
<li>链地址：将所有冲突归到一个链表上</li>
</ol>
<p>装填因子<eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mfrac><mtext>装入记录数</mtext><mtext>哈希表长度</mtext></mfrac></mrow><annotation encoding="application/x-tex">\alpha = \frac{装入记录数}{哈希表长度}</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" style="margin-right:0.0037em;">α</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:1.217331em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord cjk_fallback mtight">哈</span><span class="mord cjk_fallback mtight">希</span><span class="mord cjk_fallback mtight">表</span><span class="mord cjk_fallback mtight">长</span><span class="mord cjk_fallback mtight">度</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord cjk_fallback mtight">装</span><span class="mord cjk_fallback mtight">入</span><span class="mord cjk_fallback mtight">记</span><span class="mord cjk_fallback mtight">录</span><span class="mord cjk_fallback mtight">数</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></eq></p>
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