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.
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 simply record this forest of trees with an array p of size N items where p[i] records the parent of item i and if p[i] = i, then i is the root of this tree and also the representative item of the set that contains item i.
Once again, look at the visualization above and determine the values inside this array p.
Discuss: If i is the root of the tree that contains it, can we set p[i] = -1 instead of p[i] = i? What are the implications?
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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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.
If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.
FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.
Initialize(N, M): Create N items and form M disjoint sets with these N items. We randomly pick two disjoint sets and merge them until we have M random disjoint sets. Currently this setup is not random enough, i.e., it cannot create tall trees for example.
The default form is Initialize(N, N), i.e., M = N, all with p[i] = i and rank[i] = 0 (all these rank values are initially not shown). The time complexity of this operation is clearly O(N).
Due to the limitation of screen size, we set 1 ≤ N ≤ 16.
If we execute FindSet(12), we will immediately get vertex 12. If we execute FindSet(9) we will get vertex 6 after 1 step and no other change.
Now try executing FindSet(0). If this is your first call on this default UFDS example, it will return vertex 3 after 2 steps and then modify the underlying UFDS structure due to path-compression in action (that is, vertex 0 points to vertex 3 directly). Notice that rank value of rank[1] = 1 is now wrong as vertex 1 becomes a new leaf. However, we will not bother to update its value.
Notice that the next time you execute FindSet(0) again, it will be (much) faster as the path has been compressed. For now, we assume that FindSet(i) runs in O(1).
UnionSet(i, j): If item i and j come from two disjoint sets initially, we link the representative item of the shorter tree/disjoint set to the representative item of the taller tree/disjoint set (otherwise, we do nothing). This is also done in O(1).
This is union-by-rank heuristic in action and will cause the resulting tree to be relatively short. Only if the two trees are equally tall before union (by comparing their rank values heuristically — note that we are not comparing their actual — the current — heights), then the rank of the resulting tree will increase by one unit.
On the same default example, try UnionSet(9, 12). As the tree that represents disjoint set {6, 9} is currently taller (according to the value of rank[6] = 1), then the shorter tree that represents disjoint set {12} will be slotted under vertex 6, without increasing the height of the combined tree at all.
On the same default example, try UnionSet(0, 11). Notice that the ranks of vertex 3 and vertex 5 are the same rank[3] = rank[5] = 2. Thus, we can either put vertex 3 under vertex 5 (our implementation) or vertex 5 under vertex 3 (both will increase the resulting height of the combined tree by 1). Notice the indirect path-compression heuristic in action.
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Quiz: Starting with N=8 disjoint sets, how tall (heuristically) can the resulting final tree if we call 7 UnionSet(i, j) operations strategically?
Quiz: Starting with N=8 disjoint sets, how short (heuristically) can the resulting final tree if we call 7 UnionSet(i, j) operations strategically?
讨论：为什么？
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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.
If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.
FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.
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So far, we say that FindSet(i), IsSameSet(i, j), and UnionSet(i, j) runs in O(1). Actually they run in O(α(N)) if the UFDS is implemented with both path-compression and union-by-rank heuristics. The analysis is quite involved and is skipped in this visualization.
This α(N) is called the inverse Ackermann function that grows extremely slowly. For practical usage of this UFDS data structure (assuming N ≤ 1M), we have α(1M) ≈ 1.
The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.
If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.
FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.
The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.
If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.
FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.
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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例子
Initialise(初始化）
FindSet(查找)
IsSameSet(在同一集）
UnionSet
>
四个不交集
三个不交集
GtCP 7.41
Default
Two disjoint sets
CLRS3 21.4
CP4 2.11
等级3
等级2
等级1
1棵等级4的树
N =
into
M =
disjoint sets of rank ≤ 1
执行
i =
执行
i =
j =
执行
i =
j =
执行
We use cookies to improve our website. By clicking ACCEPT, you agree to our use of Google Analytics for analysing user behaviour and improving user experience as described in our Privacy Policy. By clicking reject, only cookies necessary for site functions will be used.
Initially conceived in 2011 by Dr. 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:
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
List of translators who have contributed ≥100 translations can be found at statistics page.
致谢
The birth of this project was made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).
使用条款
✕
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. 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 courses 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 course lecturer to keep track of your e-lecture slides reading and online quiz training progresses that is needed to run the course 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 course 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.
