algorithm/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2025-07-21 23:16:41 +08:00
Code Issues Packages Projects Releases Wiki Activity

deploy

Browse Source
This commit is contained in:
krahets
2023-08-20 13:37:20 +08:00
parent 88e0b11361
commit 96fded547b
35 changed files with 777 additions and 716 deletions
Download Patch File Download Diff File Expand all files Collapse all files

50
chapter_computational_complexity/space_complexity/index.html
View File

@ -682,36 +682,36 @@
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#o1" class="md-nav__link">
常数阶 \(O(1)\)
<a href="#1-o1" class="md-nav__link">
1. &nbsp; 常数阶 \(O(1)\)
</a>
</li>
<li class="md-nav__item">
<a href="#on" class="md-nav__link">
线性阶 \(O(n)\)
<a href="#2-on" class="md-nav__link">
2. &nbsp; 线性阶 \(O(n)\)
</a>
</li>
<li class="md-nav__item">
<a href="#on2" class="md-nav__link">
平方阶 \(O(n^2)\)
<a href="#3-on2" class="md-nav__link">
3. &nbsp; 平方阶 \(O(n^2)\)
</a>
</li>
<li class="md-nav__item">
<a href="#o2n" class="md-nav__link">
指数阶 \(O(2^n)\)
<a href="#4-o2n" class="md-nav__link">
4. &nbsp; 指数阶 \(O(2^n)\)
</a>
</li>
<li class="md-nav__item">
<a href="#olog-n" class="md-nav__link">
对数阶 \(O(\log n)\)
<a href="#5-olog-n" class="md-nav__link">
5. &nbsp; 对数阶 \(O(\log n)\)
</a>
</li>
@ -3452,36 +3452,36 @@
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#o1" class="md-nav__link">
常数阶 \(O(1)\)
<a href="#1-o1" class="md-nav__link">
1. &nbsp; 常数阶 \(O(1)\)
</a>
</li>
<li class="md-nav__item">
<a href="#on" class="md-nav__link">
线性阶 \(O(n)\)
<a href="#2-on" class="md-nav__link">
2. &nbsp; 线性阶 \(O(n)\)
</a>
</li>
<li class="md-nav__item">
<a href="#on2" class="md-nav__link">
平方阶 \(O(n^2)\)
<a href="#3-on2" class="md-nav__link">
3. &nbsp; 平方阶 \(O(n^2)\)
</a>
</li>
<li class="md-nav__item">
<a href="#o2n" class="md-nav__link">
指数阶 \(O(2^n)\)
<a href="#4-o2n" class="md-nav__link">
4. &nbsp; 指数阶 \(O(2^n)\)
</a>
</li>
<li class="md-nav__item">
<a href="#olog-n" class="md-nav__link">
对数阶 \(O(\log n)\)
<a href="#5-olog-n" class="md-nav__link">
5. &nbsp; 对数阶 \(O(\log n)\)
</a>
</li>
@ -4121,7 +4121,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p class="admonition-title">Tip</p>
<p>部分示例代码需要一些前置知识，包括数组、链表、二叉树、递归算法等。如果你遇到看不懂的地方，可以在学习完后面章节后再来复习。</p>
</div>
<h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">&para;</a></h3>
<h3 id="1-o1">1. &nbsp; 常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-o1" title="Permanent link">&para;</a></h3>
<p>常数阶常见于数量与输入数据大小 <span class="arithmatex">\(n\)</span> 无关的常量、变量、对象。</p>
<p>需要注意的是，在循环中初始化变量或调用函数而占用的内存，在进入下一循环后就会被释放，即不会累积占用空间，空间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JS</label><label for="__tabbed_4_6">TS</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
@ -4431,7 +4431,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
</div>
</div>
</div>
<h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">&para;</a></h3>
<h3 id="2-on">2. &nbsp; 线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-on" title="Permanent link">&para;</a></h3>
<p>线性阶常见于元素数量与 <span class="arithmatex">\(n\)</span> 成正比的数组、链表、栈、队列等。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
<div class="tabbed-content">
@ -4798,7 +4798,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><img alt="递归函数产生的线性阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></p>
<p align="center"> 图：递归函数产生的线性阶空间复杂度 </p>
<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">&para;</a></h3>
<h3 id="3-on2">3. &nbsp; 平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-on2" title="Permanent link">&para;</a></h3>
<p>平方阶常见于矩阵和图，元素数量与 <span class="arithmatex">\(n\)</span> 成平方关系。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
<div class="tabbed-content">
@ -5133,7 +5133,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><img alt="递归函数产生的平方阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></p>
<p align="center"> 图：递归函数产生的平方阶空间复杂度 </p>
<h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">&para;</a></h3>
<h3 id="4-o2n">4. &nbsp; 指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-o2n" title="Permanent link">&para;</a></h3>
<p>指数阶常见于二叉树。高度为 <span class="arithmatex">\(n\)</span> 的「满二叉树」的节点数量为 <span class="arithmatex">\(2^n - 1\)</span> ，占用 <span class="arithmatex">\(O(2^n)\)</span> 空间。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="9:12"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JS</label><label for="__tabbed_9_6">TS</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label><label for="__tabbed_9_9">Swift</label><label for="__tabbed_9_10">Zig</label><label for="__tabbed_9_11">Dart</label><label for="__tabbed_9_12">Rust</label></div>
<div class="tabbed-content">
@ -5282,7 +5282,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p><img alt="满二叉树产生的指数阶空间复杂度" src="../space_complexity.assets/space_complexity_exponential.png" /></p>
<p align="center"> 图：满二叉树产生的指数阶空间复杂度 </p>
<h3 id="olog-n">对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#olog-n" title="Permanent link">&para;</a></h3>
<h3 id="5-olog-n">5. &nbsp; 对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-olog-n" title="Permanent link">&para;</a></h3>
<p>对数阶常见于分治算法和数据类型转换等。</p>
<p>例如“归并排序”算法，输入长度为 <span class="arithmatex">\(n\)</span> 的数组，每轮递归将数组从中点划分为两半，形成高度为 <span class="arithmatex">\(\log n\)</span> 的递归树，使用 <span class="arithmatex">\(O(\log n)\)</span> 栈帧空间。</p>
<p>再例如“数字转化为字符串”，输入任意正整数 <span class="arithmatex">\(n\)</span> ，它的位数为 <span class="arithmatex">\(\log_{10} n\)</span> ，即对应字符串长度为 <span class="arithmatex">\(\log_{10} n\)</span> ，因此空间复杂度为 <span class="arithmatex">\(O(\log_{10} n) = O(\log n)\)</span> 。</p>