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Bitmask provide an efficient way to manipulate a small set of Booleans that is stored as a 32-(or 64-)bit signed integer in base-10 but interpreted as a short 32-(or 64-) characters string.

By using bitwise operations, each bit of the integer can be checked, turned on (or turned off) easily and quickly. It can be used in various algorithms such as the Dynamic Programming solution for Travelling Salesperson Problem to speed up crucial (small) set-based operations.

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If you are an NUS student and a repeat visitor, please login.

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The visualisation of a bitmask data structure and its operations are shown above.

The top row describes the indices of the bits of the input integer S starting from the 0-th bit as the rightmost bit. Note that we use 1-digit hexadecimal notation of A/B/../E for the 10/11/../14-th bit.

The second row describes the bits of the input integer S. In practical applications, this is usually a 32-bit signed integer (e.g., int in C++/most other programming languages in 2023). Sometimes, this is a 64-bit signed integer (e.g., long long in C++). For this visualization, we limit S to 16-bit signed integer.

The third row describes the bits of the (bit)mask that will be applied to S together with the associated bitwise operation.

The last row describes the result.

Pro-tip 1: Since you are not logged-in, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown]/[PageUp] to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use [→ or ↓/← or ↑] to do the same),and [Esc] to toggle between this e-Lecture mode and exploration mode.

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In this visualization, we currently support 6 Bitmask operations (all run in O(1)):

1. Set S (several ways)
2. Set j-th Bit
3. Check j-th Bit
4. Clear j-th Bit
5. Toggle j-th Bit
6. Least Significant Bit

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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We can enter a (small) Integer between [0..32767 (215-1)] in Decimal (base-10) and the Binary (base-2) form of S will be visualized.

Alternatively, we can select any random Integer between the same range [0..32767], a random powers of 2 (with specific pattern of '10...0'), or a random powers of 2 minus 1 (with specific pattern of '11...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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We can enter the index of the j-th Bit of S to be set (turned on).

Note that index starts from 0 but counted from the rightmost bit (refer to the top row).

The bitwise operation is simple: S OR (1 << j).

Setting a bit that is already on will not change anything.

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We can enter the index of the j-th Bit of S to be checked (on whether it is on or off).

Note that index starts from 0 but counted from the rightmost bit (refer to the top row).

The bitwise operation is simple: S AND (1 << j).

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We can enter the index of the j-th Bit of S to be cleared (turned off).

Note that index starts from 0 but counted from the rightmost bit (refer to the top row).

The bitwise operation is simple: S AND ~(1 << j).

Clearing a bit that is already off will not change anything.

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We can enter the index of the j-th Bit of S to be toggled (on → off or off → on).

Note that index starts from 0 but counted from the rightmost bit (refer to the top row).

The bitwise operation is simple: S XOR (1 << j).

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This operation requires no input. It is a special operation to quickly identify the rightmost bit that is on in S.

The bitwise operation is simple: S AND (-S).

Note that in Two's complement, -S = NOT(S)+1.

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For source code example involving bitmask/bit manipulation, please review: bit_manipulation.cpp | py | java | ml.

And for more challenging problems involving bitmask/bit manipulation, try the following online judge problems: UVa 11933 - Splitting Numbers and Kattis - bitbybit.

Beyond these simple applications, bitmask frequently used as low-level optimizations in a few advanced algorithms, so get ready when you encounter bitmask as sub-component of the bigger algorithms.

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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Increment

Set S

Set j-th Bit

Check j-th Bit

Toggle j-th Bit

Least Significant Bit

>
S =

Go

Random

Random 2^N

Random 2^N-1

j =

Go

j =

Go

j =

Go

j =

Go

Initially conceived in 2011 by Associate Professor Steven Halim, VisuAlgo aimed to facilitate a deeper understanding of data structures and algorithms for his students by providing a self-paced, interactive learning platform.

Featuring numerous advanced algorithms discussed in Dr. Steven Halim's book, 'Competitive Programming' — co-authored with Dr. Felix Halim and Dr. Suhendry Effendy — VisuAlgo remains the exclusive platform for visualizing and animating several of these complex algorithms even after a decade.

While primarily designed for National University of Singapore (NUS) students enrolled in various data structure and algorithm courses (e.g., CS1010/equivalent, CS2040/equivalent (including IT5003), CS3230, CS3233, and CS4234), VisuAlgo also serves as a valuable resource for inquisitive minds worldwide, promoting online learning.

Initially, VisuAlgo was not designed for small touch screens like smartphones, as intricate algorithm visualizations required substantial pixel space and click-and-drag interactions. For an optimal user experience, a minimum screen resolution of 1366x768 is recommended. However, since April 2022, a mobile (lite) version of VisuAlgo has been made available, making it possible to use a subset of VisuAlgo features on smartphone screens.

VisuAlgo remains a work in progress, with the ongoing development of more complex visualizations. At present, the platform features 24 visualization modules.

Equipped with a built-in question generator and answer verifier, VisuAlgo's "online quiz system" enables students to test their knowledge of basic data structures and algorithms. Questions are randomly generated based on specific rules, and students' answers are automatically graded upon submission to our grading server. As more CS instructors adopt this online quiz system worldwide, it could effectively eliminate manual basic data structure and algorithm questions from standard Computer Science exams in many universities. By assigning a small (but non-zero) weight to passing the online quiz, CS instructors can significantly enhance their students' mastery of these basic concepts, as they have access to an almost unlimited number of practice questions that can be instantly verified before taking the online quiz. Each VisuAlgo visualization module now includes its own online quiz component.

VisuAlgo has been translated into three primary languages: English, Chinese, and Indonesian. Additionally, we have authored public notes about VisuAlgo in various languages, including Indonesian, Korean, Vietnamese, and Thai:

id, kr, vn, th.

#### Team

Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

Final Year Project/UROP students 1
Jul 2012-Dec 2013: Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy
Jun 2013-Apr 2014 Rose Marie Tan Zhao Yun, Ivan Reinaldo

CDTL TEG 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

Final Year Project/UROP students 2
Jun 2014-Apr 2015: Erin Teo Yi Ling, Wang Zi
Jun 2016-Dec 2017: Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir
Aug 2021-Apr 2023: Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long, Ting Xiao, Lim Dewen Aloysius

Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Final Year Project/UROP students 3
Aug 2023-Apr 2024: Xiong Jingya, Radian Krisno, Ng Wee Han

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

Acknowledgements
NUS CDTL gave Teaching Enhancement Grant to kickstart this project.

For Academic Year 2023/24, a generous donation from Optiver will be used to further develop VisuAlgo.

VisuAlgo is generously offered at no cost to the global Computer Science community. If you appreciate VisuAlgo, we kindly request that you spread the word about its existence to fellow Computer Science students and instructors. You can share VisuAlgo through social media platforms (e.g., Facebook, YouTube, Instagram, TikTok, Twitter, etc), course webpages, blog reviews, emails, and more.

Data Structures and Algorithms (DSA) students and instructors are welcome to use this website directly for their classes. If you capture screenshots or videos from this site, feel free to use them elsewhere, provided that you cite the URL of this website (https://visualgo.net) and/or the list of publications below as references. However, please refrain from downloading VisuAlgo's client-side files and hosting them on your website, as this constitutes plagiarism. At this time, we do not permit others to fork this project or create VisuAlgo variants. Personal use of an offline copy of the client-side VisuAlgo is acceptable.

Please note that VisuAlgo's online quiz component has a substantial server-side element, and it is not easy to save server-side scripts and databases locally. Currently, the general public can access the online quiz system only through the 'training mode.' The 'test mode' offers a more controlled environment for using randomly generated questions and automatic verification in real examinations at NUS.

List of Publications

This work has been presented 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).

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Associate Professor 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 Associate Professor Steven Halim. His contact is the concatenation of his name and add gmail dot com.