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A Matching in a graph G = (V, E) is a subset M of E edges in G such that no two of which meet at a common vertex.

Maximum Cardinality Matching (MCM) problem is a Graph Matching problem where we seek a matching M that contains the largest possible number of edges. A possible variant is Perfect Matching where all V vertices are matched, i.e., the cardinality of M is V/2.

A Bipartite Graph is a graph whose vertices can be partitioned into two disjoint sets U and V such that every edge can only connect a vertex in U to a vertex in V.

Maximum Cardinality Bipartite Matching (MCBM) problem is the MCM problem in a Bipartite Graph, which is a lot easier than MCM problem in a General Graph.

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.


Visualisasi ini saat ini terbatas pada graf-graf tidak berbobot saja. Sehingga, kita saat ini tidak mendukung varian-varian masalah Pencocokan Graf variants yang berhubungan dengan graf-graf berbobot...

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.


To switch between the unweighted MCBM (default, as it is much more popular) and unweighted MCM mode, click the respective header.

Here is an example of MCM mode. In MCM mode, one can draw a General, not necessarily Bipartite graphs. However, the graphs are unweighted (all edges have uniform weight 1).

The available algorithms are different in the two modes.

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.


You can view the visualisation here!

For Bipartite Graph visualization, we will re-layout the vertices of the graph so that the two disjoint sets (U and V) are clearly visible as Left (U) and Right (V) sets. For General Graph, we do not relayout the vertices.

Initially, edges have grey color. Matched edges will have black color. Free/Matched edges along an augmenting path will have Orange/Light Blue colors, respectively.

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.


There are three different sources for specifying an input graph:

  1. Draw Graph: You can draw any undirected unweighted graph as the input graph (note that in MCBM mode, the drawn input graph will be relayout into a nice Bipartite graph layout during algorithm animation),
  2. Modeling: A lot of graph problems can be reduced into an MCBM problem. In this visualization, we have the modeling examples for the famous Rook Attack problem and standard MCBM problem (also valid in MCM mode).
  3. Examples: You can select from the list of our example graphs to get you started. The list of examples is slightly different in the two MCBM vs MCM modes.

There are several Max Cardinality Bipartite Matching (MCBM) algorithms in this visualization, plus one more in Max Flow visualization:

  1. O(VE) Augmenting Path Algorithm (without greedy pre-processing),
  2. O(√(V)E) Dinic's Max Flow Algorithm, see Max Flow visualization, select Modeling → Bipartite Matching → All 1, then use Dinic's algorithm.
  3. O(√(V)E) Hopcroft-Karp Algorithm,
  4. O(kE) Augmenting Path Algorithm++ (with randomized greedy pre-processing),

PS1: Although possible, we will likely not use O(V3) Edmonds' Matching Algorithm if the input is guaranteed to be a Bipartite Graph.

PS2: Although possible, we will also likely not use O(V3) Kuhn-Munkres Algorithm if the input is guaranteed to be an unweighted Bipartite Graph.


Augmenting Path is a path that starts from a free (unmatched) vertex u in graph G (note that G does not necessarily has to be a bipartite graph), alternates through unmatched, matched, ..., unmatched edges in G, until it ends at another free vertex v. If we flip the edge status along that augmenting path, we will increase the number of edges in the matching set M by 1 and eliminates this augmenting path.

In 1957, Claude Berge proposes the following theorem/lemma: A matching M in graph G is maximum iff there is no more augmenting path in G.

The Augmenting Path Algorithm is a simple O(V*(V+E)) = O(V2 + VE) = O(VE) implementation of that lemma (on Bipartite Graph): Find and then eliminate augmenting paths in Bipartite Graph G. Click Augmenting Path Algorithm Demo to visualize this algorithm on the currently displayed random Bipartite Graph.

vi match, vis; // global variables

int Aug(int L) {
if (vis[L]) return 0; // L visited, return 0
vis[L] = 1;
for (auto &R : AL[L])
if ((match[R] == -1) || Aug(match[R])) {
match[R] = L; // flip status
return 1; // found 1 matching
return 0; // no matching
// in int main(), build the bipartite graph
// use directed edges from left set (of size VLeft) to right set
int MCBM = 0;
match.assign(V, -1);
for (int L = 0; L < VLeft; ++L) {
vis.assign(VLeft, 0);
MCBM += Aug(L); // find augmenting path starting from L
printf("Found %d matchings\n", MCBM);

Please see the full implementation at Competitive Programming book repository: mcbm.cpp | py | java | ml.


The MCBM problem can be modeled as a Max Flow problem. Go to Max Flow visualization page and see the flow graph modeling of MCBM problem (select Modeling → Bipartite Matching → all 1).

If we use one of the fastest Max Flow algorithm, i.e., Dinic's algorithm on this flow graph, we can find Max Flow = MCBM in O(√(V)E) time — analysis omitted for now. This allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions.


If we are given a Complete Bipartite Graph KN/2,N/2, i.e.,
V = N/2+N/2 = N and E = N/2×N/2 = N2/4 ≈ N2, then
the Augmenting Path Algorithm discussed earlier will run in O(VE) = O(N×N2) = O(N3).

This is only OK for V ∈ [400..500] in a typical 1s allowed runtime in many programming competitions.

Try executing the standard Augmenting Path Algorithm on this Extreme Test Case, which is an almost complete K5,5 Bipartite Graph.


The key idea of Hopcroft-Karp (HK) Algorithm (invented in 1973) is identical to Dinic's Max Flow Algorithm discussed earlier, i.e., prioritize shortest augmenting paths (in terms of number of edges used) first. That's it, augmenting paths with 1 edge are processed first before longer augmenting paths with 3 edges, 5 edges, 7 edges, etc (the length always increase by 2 due to the nature of augmenting path in a Bipartite Graph).

Hopcroft-Karp Algorithm has time complexity of O(√(V)E) — analysis omitted for now. This allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions — the similar range as with running Dinic's algorithm on Bipartite Matching flow graph.

Try HK Algorithm on the same Extreme Test Case earlier. You will notice that HK Algorithm can find the MCBM in a much faster time than the previous standard O(VE) Augmenting Path Algorithm.


However, we can actually make the easy-to-code Augmenting Path Algorithm discussed earlier to avoid its worst case O(VE) behavior by doing O(V+E) randomized (to avoid adversary test case) greedy pre-processing before running the actual algorithm.

This O(V+E) additional pre-processing step is simple: For every vertex on the left set, match it with a randomly chosen unmatched neighbouring vertex on the right set. This way, we eliminates many trivial (one-edge) Augmenting Paths that consist of a free vertex u, an unmatched edge (u, v), and a free vertex v.

Try Augmenting Path Algorithm++ on the same Extreme Test Case earlier. Notice that the pre-processing step already eliminates many trivial 1-edge augmenting paths, making the actual Augmenting Path Algorithm only need to do little amount of additional work.


Quite often, on randomly generated Bipartite Graph, the randomized greedy pre-processing step has cleared most of the matchings.

However, we can construct test case like: Examples: Randomized Greedy Processing Killer to make randomization as ineffective as possible. For every group of 4 vertices, there are 2 matchings. Random greedy processing has 50% chance of making mistake per group (but since each group has only short Augmenting Paths, the fixes are not 'long'). Try this Test Case with Multiple Components case to see for yourself.

The worst case time complexity is no longer O(VE) but now O(kE) where k is a small integer, much smaller than V, k can be as small as 0 and is at most V/2 (any maximal matching, as with this case, has size of at least half of the maximum matching). In our empirical experiments, we estimate k to be "about √(V)" too. This version of Augmenting Path Algorithm++ also allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions.


There are two Max Cardinality Matching (MCM) algorithms in this visualization:

  1. O(V^3) Edmonds's Matching algorithm (without greedy pre-processing),
  2. O(V^3) Edmonds's Matching algorithm (with greedy pre-processing),

In General Graph, we may have Odd-Length cycle. Augmenting Path is not well defined in such graph, hence we cannot directly implement Claude Berge's theorem/lemma like what we did with Bipartite Graph.

Jack Edmonds call a path that starts from a free vertex u, alternates between free, matched, ..., free edges, and returns to the same free vertex u as Blossom. This situation is only possible if we have Odd-Length cycle, i.e., non-Bipartite Graph. Edmonds then proposed Blossom shrinking/contraction and expansion algorithm to solve this issue.

For details on how this algorithm works, read CP4 Section 9.28.

This algorithm can be implemented in O(V^3).


As with the Augmenting Path Algorithm++ for the MCBM problem, we can also do randomized greedy pre-processing step to eliminate as many 'trivial matchings' as possible upfront. This reduces the amount of work of Edmonds' Matching Algorithm, thus resulting in a faster time complexity — analysis TBA.


Kami belum menambahkan visualisasi-visualisasi untuk varian berbobot dari masalah MCBM dan MCM (untuk pekerjaan dimasa mendatang).


You are allowed to use/modify our implementation code for Augmenting Path Algorithm++: mcbm.cpp| py | java | ml

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.


Gambar Grafik


Graf-Graf Contoh

Augmenting Path


Rook Attack

Generate Random Bipartite Graph

Undirected Max Flow Killer

House of Cards

CS4234 Tutorial 3


Randomized Greedy Processing Killer


>K5,5 (Almost)


Dengan Pemasangan Acak sebelum diproses

Hopcroft Karp

Edmonds Blossom

Edmonds Blossom + Greedy

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Tentang Tim Syarat Guna Kebijakan Privasi


VisuAlgo digagas pada tahun 2011 oleh Dr Steven Halim sebagai alat untuk membantu murid-muridnya mengerti struktur-struktur data dan algoritma-algoritma, dengan memampukan mereka untuk mempelajari dasar-dasarnya secara otodidak dan dengan kecepatan mereka sendiri.

VisuAlgo mempunya banyak algoritma-algoritma tingkat lanjut yang dibahas didalam buku Dr Steven Halim ('Competitive Programming', yang ditulis bersama adiknya Dr Felix Halim dan temannya Dr Suhendry Effendy) dan lebih lagi. Hari ini, beberapa dari visualisasi/animasi algoritma-algoritma tingkat lanjut ini hanya ditemukan di VisuAlgo.

Meskipun pada khususnya didesain untuk murid-murid National University of Singapore (NUS) yang mengambil berbagai kelas-kelas struktur data dan algoritma (contoh: CS1010/setara, CS2040/setara, CS3230, CS3233, dan CS4234), sebagai pendukung pembelajaran online, kami berharap bahwa orang-orang di berbagai belahan dunia menemukan visualisasi-visualisasi di website ini berguna bagi mereka juga.

VisuAlgo tidak didesain untuk layar sentuh kecil (seperti smartphones) dari awalnya karena kami harus membuat banyak visualisasi-visualisasi algoritma kompleks yang membutuhkan banyak pixels dan gestur klik-dan-tarik untuk interaksinya. Resolusi layar minimum untuk pengalaman pengguna yang lumayan adalah 1024x768 dan hanya halaman utama VisuAlgo yang secara relatif lebih ramah dengan layar kecil. Tetapi, kami sedang bereksperimen dengan versi mobil (kecil) dari VisuAlgo yang akan siap pada April 2022.

VisuAlgo adalah proyek yang sedang terus berlangsung dan visualisasi-visualisasi yang lebih kompleks sedang dibuat.

Perkembangan yang paling menarik adalah pembuatan pertanyaan otomatis (sistem kuis online) yang bisa dipakai oleh murid-murid untuk menguji pengetahuan mereka tentang dasar struktur-struktur data dan algoritma-algoritma. Pertanyaan-pertanyaan dibuat secara acak dengan semacam rumus dan jawaban-jawaban murid-murid dinilai secara instan setelah dikirim ke server penilai kami. Sistem kuis online ini, saat sudah diadopsi oleh banyak dosen Ilmu Komputer diseluruh dunia, seharusnya bisa menghapuskan pertanyaan-pertanyaan dasar tentang struktur data dan algoritma dari ujian-ujian di banyak Universitas. Dengan memberikan bobot kecil (tapi tidak kosong) supaya murid-murid mengerjakan kuis online ini, seorang dosen Ilmu Komputer dapat dengan signifikan meningkatkan penguasaan materi dari murid-muridnya tentang pertanyaan-pertanyaan dasar ini karena murid-murid mempunyai kesempatan untuk menjawab pertanyaan-pertanyaan ini yang bisa dinilai secara instan sebelum mereka mengambil kuis online yang resmi. Mode latihan saat ini mempunyai pertanyaan-pertanyaan untuk 12 modul visualisasi. Kami akan segera menambahkan pertanyaan-pertanyaan untuk 12 modul visualisasi yang lainnya sehingga setiap setiap modul visualisasi di VisuAlgo mempunyai komponen kuis online.

Kami telah menerjemahkan halaman-halaman VisuALgo ke tiga bahasa-bahasa utama: Inggris, Mandarin, dan Indonesia. Saat ini, kami juga telah menulis catatan-catatan publik tentang VisuAlgo dalam berbagai bahasa:

id, kr, vn, th.


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

Murid-Murid S1 Peniliti 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

Murid-Murid Proyek Tahun Terakhir/UROP 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Murid-Murid S1 Peniliti 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 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Murid-Murid Proyek Tahun Terakhir/UROP 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

Murid-Murid Proyek Tahun Terakhir/UROP 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.

Ucapan Terima Kasih
Proyek ini dimungkinkan karena Hibah Pengembangan Pengajaran dari NUS Centre for Development of Teaching and Learning (CDTL).

Syarat Guna

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.

Kebijakan Privasi

Versi 1.1 (Dimutakhirkan Jum, 14 Jan 2022).

Pemberitahuan kepada semua pengunjung: Kami saat ini menggunakan Google Analytics untuk mendapatkan pengertian garis besar tentang pengunjung-pengunjung situs kami. Kami sekarang memberikan opsi kepada pengguna untuk Menerima atau Menolak pelacak ini.

Sejak Rabu, 22 Des 2021, hanya staff-staff/murid-murid National University of Singapore (NUS) dan dosen-dosen Ilmu Komputer diluar dari NUS yang telah menulis kepada Steven dapat login ke VisuAlgo, orang-orang lain di dunia harus memakai VisuAlgo sebagai pengguna anonim yang tidak benar-benar terlacak selain apa yang dilacak oleh Google Analytics.

Untuk murid-murid NUS yang mengambil mata kuliah yang menggunakan VisuAlgo: Dengan menggunakan akun VisuAlgo (sebuah tupel dari alamat email NUS resmi, nama murid resmi NUS seperti dalam daftar kelas, dan sebuah kata sandi yang dienkripsi pada sisi server — tidak ada data personal lainnya yang disimpan), anda memberikan ijin kepada dosen modul anda untuk melacak pembacaan slide-slide kuliah maya dan kemajuan latihan kuis online yang dibutuhkan untuk menjalankan modul tersebut dengan lancar. Akun VisuAlgo anda akan juga dibutuhkan untuk mengambil kuis-kuis VisuAlgo online resmi sehingga memberikan kredensial akun anda ke orang lain untuk mengerjakan Kuis Online sebagai anda adalah pelanggaran akademis.. Akun pengguna anda akan dihapus setelah modul tersebut selesai kecuali anda memilih untuk menyimpan akun anda (OPT-IN). Akses ke basis data lengkap dari VisuAlgo (dengan kata-kata sandi terenkripsi) dibatasi kepada Steven saja.

Untuk murid-murid NUS lainnya, anda dapat mendaftarkan sendiri sebuah akun VisuAlgo (OPT-IN).

Untuk dosen-dosen Ilmu Komputer di seluruh dunia yang telah menulis kepada Steven, sebuah akun VisuAlgo (alamat email (bukan-NUS), anda dapat menggunakan nama panggilan apapun, dan kata sandi terenkripsi) dibutuhkan untuk membedakan kredensial online anda dibandingkan dengan orang-orang lain di dunia. Akun anda akan dilacak seperti seorang murid NUS biasa diatas tetapi akun anda akan mempunya fitur-fiture spesifik untuk dosen-dosen Ilmu Komputer, yaitu kemampuan untuk melihat slide-slide tersembunyi yang berisi jawaban-jawaban (menarik) dari pertanyaan-pertanyaan yang dipresentasikan di slide-slide sebelumnya sebelum slide-slide tersembunyi tersebut. Anda juga dapat mengakses setingan Susah dari Kuis-Kuis Online VisuAlgo. Anda dapat dengan bebas menggunakan materi-materia untuk memperkaya kelas-kelas struktur-struktur data dan algoritma-algoritma anda. Catatan: Mungkin ada fitur-fitur khusus tambahan untuk dosen Ilmu Komputer di masa mendatang.

Untuk siapapun dengan akun VisuAlgo, anda dapat membuang akun anda sendiri bila anda tidak mau lagi diasosiasikan dengan tool VisuAlgo ini.