位掩码（位操作）

位掩码 (Bitmask) 提供了一种有效的方法来处理一小组以32或64位有符号十进制整数形式存储却以长度32/64的字符串来解读的布尔值。

通过使用位运算，可以检查布尔值的每一位是开还是关。它可以用于各种算法，例如旅行商问题的动态编程解决方案，以加速（小型的）基于集合的关键操作。

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在这里查看位掩码的数据结构及运算的可视化！

最上面一行展示了以第 0 位为最右位的整数 S 中各位 (bit) 的索引。注意我们使用16进制符号 A/B/../E 来表示第 10/11/../14 位。

第二行表示了整数 S 中的各位 (bit)。在实际应用中，S 通常是32位的有符号整数（例如C++和大多数其他语言中的 int 类型）。有时 S 是一个64位的有符号整数（如C++中的 long long 类型）。在此可视化中，我们将 S 限定为16位的有符号整数。

第三行表示了将会在位运算中用在 S 上的位掩码的各个位 (bit)。

最后一行表示结果。

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在此可视化中，我们只是六种使用常数时间的位运算：

设置 S 的值
将第 j 位设为 on
判断第 j 位的值
将第 j 位设为 off
对第 j 位取反
最低有效位

Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2021). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this.

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我们可以输入一个在 [0..32767 (2¹⁵-1)] 区间内的小整数，然后可视化其二进制的形式。

或者我们也可以随机选一个在 [0..32767] 内的整数，或二进制为 '100...0' 的 2ⁿ，或二进制为 '11...1' 的 2ⁿ-1。

Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): Spacebar to play/pause/replay the animation, ←/→ to step the animation backwards/forwards, respectively, and -/+ to decrease/increase the animation speed, respectively.

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我们可以输入第 j 位的索引来将其设为 on。

索引从 0 开始，但是从最右边的位开始数的（见上）。

位运算比较简单：S OR (1 << j) 。

将一个已经是 on 的位设为 on 不改变任何东西。

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我们可以输入第 j 位的索引来判断其状态（on 或 off）。

索引从 0 开始，但是从最右边的位开始数的（见上）。

位运算比较简单：S AND (1 << j) 。

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我们可以输入第 j 位的索引来将其设为 off。

索引从 0 开始，但是从最右边的位开始数的（见上）。

位运算比较简单：S AND ~(1 << j) 。

将一个已经是 off 的位设为 off 不改变任何东西。

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我们可以输入要第 j 位的索引来对其取反（on → off 或 off → on）。

索引从 0 开始，但是从最右边的位开始数的（见上）。

位运算比较简单：S XOR (1 << j) 。

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这个运算不需要输入。它只是一个找到 S 中最右边的值为 on 的位的特殊运算。

位运算比较简单：S AND (-S) 。

注意在补码中，-S = NOT(S)+1 。

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如果您需要位掩码/位操作的示例源代码，请看：bit_manipulation.cpp | py | java | ml。

如需测试您对位操作的理解，可以试试 Online Quiz on bitmask topic。

如果您想要关于位掩码/位操作的更有挑战性的问题，试试这里的在线题目：UVa 11933 - Splitting Numbers 和 Kattis - bitbybit。

除了这些简单的应用，位掩码还常常在高级算法中作为低级优化出现，所以您需要做好在更大的问题中碰到位掩码组件的心理准备。

You have reached the last slide. Return to 'Exploration Mode' to start exploring!

Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

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设置 S

将第 j 位设为 on

判断第 j 位的值

将第 j 位设为 off

对第 j 位取反

最低有效位

关于团队使用条款隐私政策

关于

VisuAlgo于2011年由Steven Halim博士创建，是一个允许学生以自己的速度自学基础知识，从而更好地学习数据结构与算法的工具。
VisuAlgo包含许多高级算法，这些算法在Steven Halim博士的书（“Competitive Programming”，与他的兄弟Felix Halim博士合作）和其他书中有讨论。今天，一些高级算法的可视化/动画只能在VisuAlgo中找到。
虽然本网站是专门为新加坡国立大学（NUS）学生学习各种数据结构和算法类（例如CS1010，CS2040，CS3230，CS3233，CS4234）而设，但我们作为在线学习的倡导者，我们非常希望世界各地的好奇的头脑能发现这些非常有用的算法可视化。
VisuAlgo不是从一开始就设计为在小触摸屏（例如智能手机）上工作良好，因为为了满足许多复杂算法可视化，需要大量的像素和点击并拖动手势进行交互。为得到良好的用户体验，最低屏幕分辨率应为1024x768，并且本网站只有首页相对适合移动设备。但是，我们正在测试一个准备在2022年4月发布的移动版本。
VisuAlgo是一个正在进行的项目，更复杂的可视化仍在开发中。
最令人兴奋的发展是自动问题生成器和验证器（在线测验系统），允许学生测试他们的基本数据结构和算法的知识。这些问题是通过一些随机生成的规则，学生的答案会在提交给我们的评分服务器后立即自动分级。这个在线测验系统，当它被更多的世界各地的CS教师采用，应该能从技术上消除许多大学的典型计算机科学考试手动基本数据结构和算法问题。通过在通过在线测验时设置小（但非零）的重量，CS教练可以（显着地）增加他/她的学生掌握这些基本问题，因为学生具有几乎无限数量的可以立即被验证的训练问题他们参加在线测验。培训模式目前包含12个可视化模块的问题。我们将很快添加剩余的12个可视化模块，以便VisuAlgo中的每个可视化模块都有在线测验组件。

VisuAlgo支持三种语言：英语，中文，印尼语。目前，我们还以各种语言写了有关VisuAlgo的公共注释：

id, kr, vn, th.

团队

项目领导和顾问（2011年7月至今）
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

本科生研究人员 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

最后一年项目/ UROP学生 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

最后一年项目/ UROP学生 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

本科生研究人员 2 (May 2014-Jul 2014)
Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

最后一年项目/ UROP学生 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

最后一年项目/ UROP学生 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

最后一年项目/ UROP学生 5 (Aug 2021-Dec 2022)
Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long

List of translators who have contributed ≥100 translations can be found at statistics page.

致谢
本项目运营资金是由NUS教学与学习发展中心（CDTL）的教学增进款慷慨提供的。

使用条款

VisuAlgo is free of charge for Computer Science community on earth. If you like VisuAlgo, the only "payment" that we ask of you is for you to tell the existence of VisuAlgo to other Computer Science students/instructors that you know =) via Facebook/Twitter/Instagram/TikTok posts, course webpages, blog reviews, emails, etc.

If you are a data structure and algorithm student/instructor, you are allowed to use this website directly for your classes. If you take screen shots (videos) from this website, you can use the screen shots (videos) elsewhere as long as you cite the URL of this website (https://visualgo.net) and/or list of publications below as reference. However, you are NOT allowed to download VisuAlgo (client-side) files and host it on your own website as it is plagiarism. As of now, we do NOT allow other people to fork this project and create variants of VisuAlgo. Using the offline copy of (client-side) VisuAlgo for your personal usage is fine.

Note that VisuAlgo's online quiz component is by nature has heavy server-side component and there is no easy way to save the server-side scripts and databases locally. Currently, the general public can only use the 'training mode' to access these online quiz system. Currently the 'test mode' is a more controlled environment for using these randomly generated questions and automatic verification for real examinations in NUS.

List of Publications

This work has been presented briefly at the CLI Workshop at the ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012) and this link for the short update in 2015 (to link VisuAlgo name with the previous project).

This work is done mostly by my past students.

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Dr Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Dr Steven Halim. His contact is the concatenation of his name and add gmail dot com.

隐私政策

Version 1.1 (Updated Fri, 14 Jan 2022).

Disclosure to all visitors: We currently use Google Analytics to get an overview understanding of our site visitors. We now give option for user to Accept or Reject this tracker.

Since Wed, 22 Dec 2021, only National University of Singapore (NUS) staffs/students and approved CS lecturers outside of NUS who have written a request to Steven can login to VisuAlgo, anyone else in the world will have to use VisuAlgo as an anonymous user that is not really trackable other than what are tracked by Google Analytics.

For NUS students enrolled in modules that uses VisuAlgo: By using a VisuAlgo account (a tuple of NUS official email address, NUS official student name as in the class roster, and a password that is encrypted on the server side — no other personal data is stored), you are giving a consent for your module lecturer to keep track of your e-lecture slides reading and online quiz training progresses that is needed to run the module smoothly. Your VisuAlgo account will also be needed for taking NUS official VisuAlgo Online Quizzes and thus passing your account credentials to another person to do the Online Quiz on your behalf constitutes an academic offense. Your user account will be purged after the conclusion of the module unless you choose to keep your account (OPT-IN). Access to the full VisuAlgo database (with encrypted passwords) is limited to Steven himself.

For other NUS students, you can self-register a VisuAlgo account by yourself (OPT-IN).

For other CS lecturers worldwide who have written to Steven, a VisuAlgo account (your (non-NUS) email address, you can use any display name, and encrypted password) is needed to distinguish your online credential versus the rest of the world. Your account will be tracked similarly as a normal NUS student account above but it will have CS lecturer specific features, namely the ability to see the hidden slides that contain (interesting) answers to the questions presented in the preceding slides before the hidden slides. You can also access Hard setting of the VisuAlgo Online Quizzes. You can freely use the material to enhance your data structures and algorithm classes. Note that there can be other CS lecturer specific features in the future.

For anyone with VisuAlgo account, you can remove your own account by yourself should you wish to no longer be associated with VisuAlgo tool.