Visualization of one of the simplest data structure in Computer Science: Array (and its sorted form) surprisingly has not been done in VisuAlgo since its inception 2011-January 2024...
Stay tuned while we improve this page and its features.
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(Compact) Array is among the easiest and the most versatile data structure in Computer Science. Array is built-in almost all programming languages, e.g., C++, Python ('array' is called as 'list' in Python), Java, etc.
We can use (Compact) Array to implement List ADT.
We can use (Compact) Array to solve many classic problems. When not being used as a List ADT implementation (where positional order matters), it is often beneficial to first sort the elements first so that we can utilize faster algorithms.
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(Compact) Array is a good candidate for implementing the List ADT as it is a simple construct to handle a collection of items.
When we say compact array, we mean an array that has no gap, i.e., if there are N items in the array (that has size M, where M ≥ N), then only index [0..N-1] are occupied and other indices [N..M-1] should remain empty.
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Let the compact array name be A with index [0..N-1] occupied with the items of the list.
get(i), just return A[i]. This simple operation will be unnecessarily complicated if the array is not compact.
search(v), we check each index i ∈ [0..N-1] one by one to see if A[i] == v. This is because v (if it exists) can be anywhere in index [0..N-1]. Since this visualization only accept distinct items, v can only be found at most once. In a general List ADT, we may want to have search(v) returns a list of indices.
insert(i, v), we shift items ∈ [i..N-1] to [i+1..N] (from backwards) and set A[i] = v. This is so that v is inserted correctly at index i and maintain compactness.
remove(i), we shift items ∈ [i+1..N-1] to [i..N-2], overwriting the old A[i]. This is to maintain compactness.
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get(i) is very fast: Just one access, O(1). Another CS course: 'Computer Organisation' discusses the details on this O(1) performance of this array indexing operation.
search(v) In the best case, v is found at the first position, O(1). In the worst case, v is not found in the list and we require O(N) scan to determine that.
insert(i, v) In the best case, insert at i = N, there is no shifting of element, O(1). In the worst case, insert at i = 0, we shift all N elements, O(N).
remove(i) In the best case, remove at i = N-1, there is no shifting of element, O(1). In the worst case, remove at i = 0, we shift all N elements, O(N).
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The size of the compact array M is not infinite, but a finite number. This poses a problem as the maximum size may not be known in advance in many applications.
If M is too big, then the unused spaces are wasted. If M is too small, then we will run out of space easily.
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Solution: Make M a variable. So when the array is full, we create a larger array (usually two times larger) and move the elements from the old array to the new array. Thus, there is no more limits on size other than the (usually large) physical computer memory size.
C++ STL std::vector, Python list, Java Vector, or Java ArrayList all implement this variable-size array. Note that Python list and Java ArrayList are not Linked Lists, but are actually variable-size arrays. This array visualization implements this doubling-when-full strategy.
However, the classic array-based issues of space wastage and copying/shifting items overhead are still problematic.
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There are various applications that can be done on a Compact (Integer) Array A:
Searching for a specific value v in array A,
Finding the min/max or the k-th smallest/largest value in (static) array A,
Testing for uniqueness and deleting duplicates in array A,
Counting how many time a specific value v appear in array A,
Set intersection/union between array A and another sorted array B,
Finding a target pair x ∈ A and y ∈ A such that x+y equals to a target z,
Counting how many values in array A is inside range [lo..hi], etc.
We will outline the possible actions that you can do in this page. For now, just try to guess based on the name of the function.
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We will talk about the two modes: array (the content can be unsorted) versus sorted array (the content must always be sorted, without loss of generality: sorted in non-decreasing order).
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There are already lots of (simple) applications that we can do with unsorted array.
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We can use O(N) linear search (leftmost to rightmost or vice versa) to find v,
For min/max, we can use O(N) linear search again; for k-th smallest/largest, we may need to use O(kN) algorithm1,
We can use O(N^2) nested-loop to see if any two indices in A are the same,
There are better ways, especially if the array if sorted.
1There is a faster expected O(N) QuickSelect or O(N) worst-case linear time selection.
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When the array is sorted, we open up a lot of possibilities.
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We can use O(log N) binary search on a sorted array,
A[0]/A[k-1]/A[N-k]/A[N-1] are the min/k-th smallest/k-th largest/max value in (static sorted) array A,
Duplicates, if any, will be next to each other in a sorted array A,
Same as above,
We can use modifications of merge routine of Merge Sort,
We can use two pointers method,
The index of y - the index of x + 1 (use two binary searches).
There can be other ways.
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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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:
Project Leader & Advisor (Jul 2011-present) Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS) Dr Felix Halim, Senior Software Engineer, Google (Mountain View)
Undergraduate Student Researchers 1 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 2014Rose Marie Tan Zhao Yun, Ivan Reinaldo
Undergraduate Student Researchers 2 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
Undergraduate Student Researchers 3 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, Tan Chee Heng Aug 2024-Apr 2025: Edbert Geraldy Cangdinata, Huang Xing Chen, Nicholas Patrick
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.
Terms of use
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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.
Privacy Policy
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Version 1.2 (Updated Fri, 18 Aug 2023).
Since Fri, 18 Aug 2023, we no longer use Google Analytics. Thus, all cookies that we use now are solely for the operations of this website. The annoying cookie-consent popup is now turned off even for first-time visitors.
Since Fri, 07 Jun 2023, thanks to a generous donation by Optiver, anyone in the world can self-create a VisuAlgo account to store a few customization settings (e.g., layout mode, default language, playback speed, etc).
Additionally, for NUS students, by using a VisuAlgo account (a tuple of NUS official email address, 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 Prof Halim himself.
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 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.
