>

>
1x
go to beginning previous frame pause play next frame go to end

The Union-Find Disjoint Sets (UFDS) data structure is used to model a collection of disjoint sets, which is able to efficiently (i.e., in nearly constant time) determine which set an item belongs to, test if two items belong to the same set, and union two disjoint sets into one when needed. It can be used to find connected components in an undirected graph, and can hence be used as part of Kruskal's algorithm for the Minimum Spanning Tree (MST) problem.


Note that this data structure has another alternative name: Disjoint Sets Union (DSU).


Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor.
If you are an NUS student and a repeat visitor, please login.

🕑

Lihatlah visualisasi dari contoh UFDS (Himpunan Lepas) di sini!


Setiap pohon melambangkan sebuah himpunan lepas (maka sebuah kumpulan dari himpunan-himpunan lepas tersebut membentuk sebuah hutan) dan akar (root) dari setiap pohon adalah item representatif dari himpunan lepas ini.


Sekarang berhenti dan lihatlah pohon-pohon yang sekarang sedang divisualisasikan. Ada berapa jumlah item-item disana? Berapa jumlah himpunan lepas di sana? Siapa saja anggota dari setiap himpunan lepas tersebut? Apakah item representasi dari setiap himpunan lepas yang ada?


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.

🕑
Karena kami menetapkan contoh default untuk kuliah maya ini, jawaban-jawaban anda harusnya: N=13 dan ada 4 himpunan lepas: {0,1,2,3,4,10}, {5,7,8,11}, {6,9}, {12} dengan anggota-anggota yang digaris bawahi adalah item-item representatif (dari himpunan lepas mereka).

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.

🕑

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.

🕑

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.

🕑

Pada contoh tetap yang sama, jawaban-jawaban anda harusnya p = [1, 3, 3, 3, 3, 5, 6, 5, 5, 6, 4, 8,12] dengan ukuran N = 13 untuk p[0] sampai p[12].


Anda dapat mengecek bahwa p[3] = 3p[5] = 5p[6] = 6, dan p[12] = 12, yang adalah konsisten dengan fakta bahwa {3,5,6,12} adalah item-item representatif (dari himpunan lepas mereka sendiri).

🕑

Kita juga menyimpan satu lagi informasi di dalam larik rank juga dengan ukuran N. Nilai dari rank[i] adalah batas-atas dari tinggi sub-pohon yang berakar pada simpul i yang akan digunakan sebagai heuristik pembimbing untuk operasi UnionSet(i, j). Anda akan menyadari nanti bahwa setelah heuristik 'kompresi-jalur' (akan dijelaskan segera) mengkompres sebuah jalur, nilai-nilai peringkat tidak lagi merefleksikan tinggi sesungguhnya dari sub-pohon tersebut.


Karena banyak item-item dengan peringkat 0, kami mengatur visualisasi sebagai berikut untuk mengurangi kekacauan: Hanya jika peringkat dari sebuah simpul i lebih besar dari 0, maka VisuAlgo akan menunjukkan nilai dari rank[i] (disingkat sebagai satu karakter r) sebagai teks berwarna merah dibawah simpul i.

🕑

Pada contoh tetap yang sama, verifikasi bahwa {1,4,6,8} memiliki peringkat 1 dan {3,5} memiliki peringkat 2, dan yang lainnya memiliki peringkat 0 (tidak ditunjukkan).


Pada saat ini, semua nilai-nilai peringkat adalah benar, yaitu mereka benar-benar mendeskripsikan tinggi dari sub-pohon yang berakar pada simpul tersebut. Kita akan segera melihat bahwa mereka tidak akan selalu benar di beberapa slide-slide berikutnya.

🕑

Terdapat lima operasi-operasi UFDS (Himpunan Lepas) dalam halaman visualisasi ini:
Contoh-Contoh, Inisialisasi(N), FindSet(i), IsSameSet(i, j), dan UnionSet(i, j).


Operasi pertama (Contoh-Contoh) adalah sederhana: Berikan daftar struktur-struktur Himpunan Lepas dengan berbagai karakteristik-karakteristik untuk titik permulaan anda. Mode kuliah maya ini selalu menggunakan contoh 'Empat Himpunan Lepas' sebagai titik permulaan.


Juga sadari bahwa tidak ada satupun dari contoh-contoh yang memiliki 'pohon yang tinggi'. Anda akan segera mengerti alasannya setelah kami menjelaskan dua heuristik-heuristik yang dipakai.

🕑

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. Due to the union-by-rank heuristics used and the randomness, it is very unlikely that this initialization process creates a tall tree.


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 ≤ 32. Obviously MN.

🕑

FindSet(i): Dari simpul i, pergi ke arah atas di dalam pohon secara rekursif. Yaitu, dari simpul i, kita pergi ke simpul p[i]) hingga kita sampai pada akar dari pohon tersebut, yang adalah item representasi dengan p[i] = i dari himpunan lepas ini.


Dalam operasi FindSet(i), kami menggunakan heuristik kompresi-jalur setelah setiap panggilan kepada FindSet(i) karena sekarang setiap simpul yang terdapat dalam jalur dari simpul i ke akar dari pohon ini mengetahui bahwa akar tersebut adalah item representatif mereka dan dapat langsung menunjuk kepada akar tersebut secara langsung dalam O(1).

🕑

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).

🕑

IsSameSet(i, j): Cek saja apakah FindSet(i) == FindSet(j). Fungsi ini digunakan secara ektensif pada algoritma MST Kruskal. Karena fungsi ini hanya memanggil operasi FindSet dua kali, kita akan mengasumsikan bahwa fungsi ini juga berjalan dalam O(1).


Perlu diingat bahwa fungsi FindSet dipanggil di dalam fungsi isSameSet, maka heuristik kompresi-jalur juga digunakan secara tidak langsung.

🕑

Jika kita memanggil IsSameSet(3, 5), kita harusnya mendapatkan false karena simpul 3 dan simpul 5 adalah item-item representatif dari himpunan-himpunan lepas mereka dan mereka berbeda.


Sekarang cobalah IsSameSet(0, 11) padah contoh default yang sama untuk melihat kompresi-jalur secara tidak langsung pada simpul 0 dan simpul 11. Kita harusnya mendapatkan false karena dua item-item representatif: simpul 3 dan simpul 5, adalah berbeda. Sadari bahwa nilai-nilai peringkat pada simpul {1, 5, 8} sekarang semuanya salah. Tetapi kita tidak akan memperbaikinya.

🕑

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.

🕑
Catat juga bahwa fungsi FindSet dipanggil dari fungsi UnionSet, jadi heuristik kompresi-jalur juga secara tidak langsung dipakai. Setiap kali heuristik kompresi-jalur mengkompres sebuah jalur, setidaknya satu dari nilai peringkat akan menjadi salah. Kita tidak perlu memperbaiki nilai-nilai peringkat ini karena mereka hanya dipakai sebagai heuristik pembimbing untuk fungsi UnionSet ini.
🕑

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.

🕑

Quiz: Starting with N=8 disjoint sets, how tall (heuristically) can the resulting final tree if we call 7 UnionSet(i, j) operations strategically?

rank:4
rank:1
rank:5
rank:3
rank:2

Quiz: Starting with N=8 disjoint sets, how short (heuristically) can the resulting final tree if we call 7 UnionSet(i, j) operations strategically?

rank:3
rank:2
rank:1
rank:5
rank:4


Diskusi: Kenapa?

🕑

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.

🕑

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.

🕑
Anda telah mencapai akhir dari informasi mendasar mengenai struktur data Himpunan Lepas dan kami mendorong anda untuk pergi ke Mode Eksplorasi dan mengeksplorasi struktur data mudah tapi menarik ini menggunakan contoh-contoh anda sendiri.

Akan tetapi, kami masih memiliki tantangan-tantangan Himpunan Lepas yang lebih menarik untuk anda.
🕑
Lihatlah implementasi-implementasi dari struktur data Himpunan Lepas ini dalam bahasa C++/Python/Java/OCaml dalam format Pemograman Berorientasi Objek (OOP)unionfind_ds.cpp | py | java | ml).

Anda bebas memodifikasi implementasi ini sesuai dengan kebutuhan anda karena beberapa soal-soal yang lebih sulit memerlukan pengubahan atas implementasi dasar ini.

Saya berharap suatu hari C++/Python/Java/OCaml/bahasa-bahasa pemrograman lainnya akan memasukkan struktur data menarik ini ke Java akan memasukkan struktur data menarik ini dalam perpustakaan dasar mereka.
🕑

For a few more interesting questions about this data structure, please practice on Union-Find Disjoint Sets training module.

🕑

Even after clearing the Online Quiz of this UFDS module, do you think that you have really mastered this data structure?


Let us challenge you by asking you to solve two programming problems that somewhat requires the usage of this Union-Find Disjoint Sets data structure: UVa 01329 - Corporative Network and Kattis - control.


Beware that both problems are actual International Collegiate Programming Contest (ICPC) problems, i.e., they are "not trivial".

🕑

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.

🕑

Notice that there is no 'undo' operation for Union-Find Disjoint Sets (UFDS) data structure. Once two initially disjoint sets were union-ed, it is not easy to split them back into the original two disjoint sets, especially when path compressions have flattened the combined tree.


Discussion: So what to do if we need this 'de-Union' or 'split' or 'cut' operation?

🕑

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.

🕑

Contoh-contoh

Inisialisasi

FindSet

IsSameSet

UnionSet

>

4 sets

3 sets

Two disjoint sets

1 Pohon dengan Rank 4

N =

items into

M =

disjoint sets

Lakukan

i =

Lakukan

i =
j =

Lakukan

i =
j =

Lakukan

Tentang Tim Syarat Guna Kebijakan Privasi

Tentang

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.

Tim

Pemimpin & Penasihat Proyek (Jul 2011-sekarang)
Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

Murid-Murid S1 Peniliti 1
CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

Murid-Murid Proyek Tahun Terakhir/UROP 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

Murid-Murid S1 Peniliti 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

Murid-Murid Proyek Tahun Terakhir/UROP 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

Murid-Murid S1 Peniliti 3
Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Murid-Murid Proyek Tahun Terakhir/UROP 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.

Ucapan Terima Kasih
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.

Syarat Guna

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.

Kebijakan Privasi

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.