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fa-fw post-meta-icon"></i><span class="post-meta-label">Updated</span><time class="post-meta-date-updated" datetime="2023-12-02T14:03:02.227Z" title="Updated 2023-12-02 22:03:02">2023-12-02</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/Data-Structures-and-Algorithms/">Data Structures and Algorithms</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-wordcount"><i class="far fa-file-word fa-fw post-meta-icon"></i><span class="post-meta-label">Word count:</span><span class="word-count">1k</span><span class="post-meta-separator">|</span><i class="far fa-clock fa-fw post-meta-icon"></i><span class="post-meta-label">Reading time:</span><span>3min</span></span><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="Bit Arithmetic"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">Post View:</span><span id="busuanzi_value_page_pv"></span></span><span class="post-meta-separator">|</span><span class="post-meta-commentcount"><i class="far fa-comments fa-fw post-meta-icon"></i><span class="post-meta-label">Comments:</span><a href="/2023/03/27/Bit-Arithmetic/#post-comment"><span class="gitalk-comment-count"></span></a></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="位运算"><a href="#位运算" class="headerlink" title="位运算"></a>位运算</h1><ul>
<li><strong>简介</strong>:程序中的所有数在计算机内存中都是以二进制的形式储存的。位运算就是直接对整数在内存中的二进制位进行操作。常见的运算符有与(&)、或(|)、异或(^)、取反(~)、左移(<<)、右移(>>是带符号右移 >>>无符号右移动)。</li>
<li><strong>优点</strong>:可以直接对二进制数进行操作,执行速度比较快,而且可以节省内存空间,特别是在处理大规模数据时,优势更加明显。</li>
<li><strong>应用场景</strong>:常用于处理二进制数据,如位图压缩、哈希表实现、位集合操作等。也可以用于加密算法、图像处理、网络协议等领域。</li>
<li><strong>时间复杂度</strong>:通常是$O(1)$,因为它们直接对二进制位进行操作,而不需要循环或递归等复杂的计算过程。</li>
<li><strong>技巧</strong>:<ol>
<li>按位与&:<ul>
<li>判断一个数的奇偶性(<code>x & 1 == 0</code>表示x是偶数,<code>x & 1 == 1</code>表示x是奇数)</li>
<li>清零一个数的某一位(<code>x &= ~(1 << n)</code>表示将x的第n位清零)</li>
<li>判断一个数是否是2的幂次方。例如:<code>(x & (x - 1)) == 0</code> 表示x是2的幂次方</li>
<li>Brian Kernighan算法:消除数字n的二进制表示中的最后一位1,也即计算汉明权重(Hamming Weight):<code>n & (n-1)</code></li>
<li>取出 x 的二进制表示中最低位的1 :<code>x & -x</code></li>
</ul>
</li>
<li>按位或|:<ul>
<li>将一个数的某一位设置为1 ( <code>x |= (1 << n)</code> 表示将x的第n位设置为1)</li>
</ul>
</li>
<li>按位异或^:<ul>
<li>可以用来交换两个变量的值(<code>a ^= b; b ^= a; a ^= b;</code>)</li>
<li>检查一个数是否出现了奇数次。对于数组中出现两次的数,使用异或操作可以找到它们,任何数和其自身做异或运算,结果是0,即 <code>a ^ a = 0</code></li>
<li>任何数和 0 做异或运算,结果仍然是原来的数,即 <code>a ^ 0 = a</code></li>
<li>异或运算满足交换律和结合律,即<code>a ⊕ b ⊕ a = b ⊕ a ⊕ a = b ⊕ (a ⊕ a) = b ⊕ 0 = b</code></li>
</ul>
</li>
<li>左移位<<:可以用来计算2的幂次方(<code>1 << n</code>表示2的n次方)</li>
<li>右移位>>:可以用来计算除以2的幂次方(<code>x >> n</code>表示将x除以2的n次方)</li>
</ol>
</li>
<li><strong>常用方法</strong>:<ul>
<li><code>bin()</code>获取数字的二进制值</li>
<li><code>ord()</code>输入单个字符返回其ASCII值(0-255)或unicode值</li>
<li><code>chr()</code>输入一个整数[0,255]返回其ASCII值</li>
<li><code>int("s",x)</code>将指定进制x的字符s转换成十进制值</li>
<li>正数的反码和补码都与原码相同。负数的二进制用补码表示。负数的反码为对该数的原码除符号位外各位取反。负数的补码为对该数的原码除符号位外各位取反,然后在最后一位加1 。</li>
</ul>
</li>
</ul>
<p>Python 没有 int, long 等不同长度变量,即在编程时无变量位数的概念。</p>
<ul>
<li>获取负数的补码:需要将数字与十六进制数 0xffffffff 相与。可s理解为舍去此数字 32 位以上的数字(将 32 位以上都变为 00),从无限长度变为一个 32 位整数。</li>
<li>返回前数字还原: 若补码 aaa 为负数( 0x7fffffff 是最大的正数的补码 ),需执行 <code>~(a ^ x)</code> 操作,将补码还原至 Python 的存储格式。 a ^ x 运算将 1 至 32 位按位取反; ~ 运算是将整个数字取反;因此, ~(a ^ x) 是将 32 位以上的位取反,1 至 32 位不变</li>
</ul>
<blockquote>
<p>参考博客:</p>
<ul>
<li><a target="_blank" rel="noopener" href="https://bigsai.blog.csdn.net/article/details/113245972">位运算介绍与经典例题总结</a></li>
<li><a target="_blank" rel="noopener" href="https://blog.csdn.net/w464960660/article/details/94304069?ops_request_misc=%257B%2522request%255Fid%2522%253A%2522161197629816780266247321%2522%252C%2522scm%2522%253A%252220140713.130102334..%2522%257D&request_id=161197629816780266247321&biz_id=0&utm_medium=distribute.pc_search_result.none-task-blog-2~all~sobaiduend~default-2-94304069.first_rank_v2_pc_rank_v29_10&utm_term=%E4%BD%8D%E6%8E%A9%E7%A0%81&spm=1018.2226.3001.4187">位掩码(BitMask)</a></li>
<li><a target="_blank" rel="noopener" href="https://blog.csdn.net/mjLlm/article/details/101559522?ops_request_misc=%257B%2522request%255Fid%2522%253A%2522161214219916780299011814%2522%252C%2522scm%2522%253A%252220140713.130102334..%2522%257D&request_id=161214219916780299011814&biz_id=0&utm_medium=distribute.pc_search_result.none-task-blog-2~all~top_click~default-1-101559522.first_rank_v2_pc_rank_v29_10&utm_term=unicode&spm=1018.2226.3001.4187">Unicode详解</a></li>
</ul>
</blockquote>
<h3 id="力扣指南"><a href="#力扣指南" class="headerlink" title="力扣指南"></a>力扣指南</h3><table>
<tr>
<th align='center', colspan="3">位运算</th>
</tr>
<tr>
<th>题目</th>
<th>技巧</th>
<th>难度</th>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/add-binary/description>✅67. 二进制求和</td>
<td>简直不是人想出来的!!!</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/single-number/description>✅136. 只出现一次的数字</td>
<td>a ^ 0 = a; a ^ a = 0</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/single-number-ii/description>✅137. 只出现一次的数字 II</td>
<td>(a >> i) & 1 取出a的第i位</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/repeated-dna-sequences/description>✅187. 重复的DNA序列</td>
<td>用二进制表示x = ((x << 2) | bin[ch]) & ((1 << 20) - 1)</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/reverse-bits/description>❌✅190. 颠倒二进制位</td>
<td>还有个分治法</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/number-of-1-bits/description>✅191. 位1的个数</td>
<td>n & (n - 1)消除二进制n的最后一位1</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/bitwise-and-of-numbers-range/description>✅201. 数字范围按位与</td>
<td>寻找公共前缀,n&(n-1)消除最后一位1</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/power-of-two/description>✅231. 2 的幂</td>
<td>同191 260</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/single-number-iii/description>✅260. 只出现一次的数字 III</td>
<td>x的最后一位1:x & -x</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/missing-number/description>✅268. 丢失的数字</td>
<td>a ^ a = 0</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/find-the-duplicate-number/description>❌287. 寻找重复数</td>
<td>难以理解</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/maximum-product-of-word-lengths/description>✅318. 最大单词长度乘积</td>
<td>将字母用掩码表示,取并为0则表示二者没用重复字符</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/counting-bits/description>❌✅338. 比特位计数</td>
<td>x&(x-1)消除1,还有动态规划解法</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/power-of-four/description>✅342. 4的幂</td>
<td>4的幂是2的偶数次幂,且%3=1</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/sum-of-two-integers/description>✅371. 两整数之和</td>
<td>python整型无限长,用32位补码表示32位有符号整型,负数最高位全为1</td>
<td>🌟🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/find-the-difference/description>✅389. 找不同</td>
<td>ord(a), chr(1)</td>
<td>🌟</td>
</tr>
<tr>
<td><a href=https://leetcode.cn/problems/set-mismatch/description>✅645. 错误的集合</td>
<td>同260</td>
<td>🌟</td>
</tr>
</table>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">Author: </span><span class="post-copyright-info"><a href="mailto:undefined">贪钱算法还我头发</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">Link: </span><span class="post-copyright-info"><a href="https://xfliu1998.github.io/2023/03/27/Bit-Arithmetic/">https://xfliu1998.github.io/2023/03/27/Bit-Arithmetic/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">Copyright Notice: </span><span class="post-copyright-info">All articles in this blog are licensed under <a target="_blank" rel="noopener" href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a> unless stating additionally.</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/Data-Structures-and-Algorithms/">Data Structures and Algorithms</a><a class="post-meta__tags" 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