7    VisuAlgo    Graph Traversal (DFS/BFS)
Exploration Mode ▿

Draw Graph

Random Graph

Example Graphs

Depth-First Search

Breadth-First Search

Topological Sort

Bipartite Graph Check

Cut Vertex & Bridge

SCC Algorithms

2-SAT Checker

CP3 4.1

CP3 4.3

CP3 4.4 DAG

CP3 4.9

CP3 4.17 DAG

CP3 4.18 DAG, Bipartite

CP3 4.19 Bipartite



DFS version

BFS version (Kahn's algorithm)

DFS version

BFS version

Kosaraju's Algorithm

Tarjan's Algorithm

Number of clauses

Number of variables


About Team Terms of use
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VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace.

VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim) and beyond. Today, some of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Though specifically designed for National University of Singapore (NUS) students taking various data structure and algorithm classes (e.g. CS1010, CS1020, CS2010, CS2020, CS3230, and CS3230), as advocators of online learning, we hope that curious minds around the world will find these visualisations useful too.

VisuAlgo is not designed to work well on small touch screens (e.g. smartphones) from the outset due to the need to cater for many complex algorithm visualizations that require lots of pixels and click-and-drag gestures for interaction. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly.

VisuAlgo is an ongoing project and more complex visualisations are still being developed.

The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms. The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. This online quiz system, when it is adopted by more CS instructors worldwide, should technically eliminate manual basic data structure and algorithm questions from typical Computer Science examinations in many Universities. By setting a small (but non-zero) weightage on passing the online quiz, a CS instructor can (significantly) increase his/her students mastery on these basic questions as the students have virtually infinite number of training questions that can be verified instantly before they take the online quiz. The training mode currently contains questions for 12 visualization modules. We will soon add the remaining 8 visualization modules so that every visualization module in VisuAlgo have online quiz component.

Another active branch of development is the internationalization sub-project of VisuAlgo. We want to prepare a database of CS terminologies for all English text that ever appear in VisuAlgo system. This is a big task and requires crowdsourcing. Once the system is ready, we will invite VisuAlgo visitors to contribute, especially if you are not a native English speaker. Currently, we have also written public notes about VisuAlgo in various languages: zh, id, kr, vn, th.


Project Leader & Advisor (Jul 2011-present)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

Undergraduate Student Researchers 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

Final Year Project/UROP students 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Undergraduate Student Researchers 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 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Final Year Project/UROP students 4 (Jun 2016-Apr 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle

This project is made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).

Terms of use

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, course webpage, blog review, email, 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 (http://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 a real examination in NUS. Other interested CS instructor should contact Steven if you want to try such 'test mode'.

List of Publications

This work has been presented briefly at the CLI Workshop at the ACM 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).

This work is done mostly by my past students. The most recent final reports are here: Erin, Wang Zi, Rose, Ivan.

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.

Given a graph, we can use DFS (Depth-First-Search) or BFS (Breadth-Frist-Search) to traverse the graph and explore the features/properties of the graph. Each graph traversal algorithm has its own characteristics that we will explore in this visualization.

This visualization is rich with a lot of DFS and BFS variants such as:

  1. Topological Sort algorithm (both DFS and BFS/Kahn algorithm version),
  2. Bipartite Graph Checker algorithm (both DFS and BFS version),
  3. Cut Vertex & Bridge finding algorithm,
  4. Strongly Connected Components (SCC) finding algorithms
    (both Kosaraju and Tarjan version), and
  5. 2-SAT Checker algorithm.
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When the chosen graph traversal algorithm is running, the animation will be shown here.

We play with vertex+edge color and occassionally the extra text under the vertex (in red font) to highlight the changes.

All graph traversal algorithms work on directed graphs (this is the default setting, where each edge has an arrowtip to indicate its direction) but the Bipartite Graph Check algorithm and the Cut Vertex & Bridge finding algorithm requires the undirected graphs (the conversion is done automatically by this visualization).

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There are three different sources for specifying an input graph:

  1. Draw Graph: You can draw any unweighted directed graph as the input graph.
  2. Random Graph: You can tap into our graph database to see other random unweighted directed graph that have been drawn by other users (currently disabled).
  3. Example Graphs: You can select from the list of our selected example graphs to get you started.
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One of the most basic graph traversal algorithm is the O(V+E) Depth-First Search (DFS) algorithm. As it name implies, DFS starts from a source vertex and uses recursion (implicit stack) to order the visitation sequence as deep as possible before backtracking. DFS also uses a Boolean array to distinguish visited and unvisited vertices.

In this visualization, we augment the DFS algorithm to also include the red = tree edge, grey = forward/cross edge, and blue = back edge of the DFS spanning tree. We are particularly interested with the presence of blue = back edge as its presence shows that the traversed graph is cyclic while its absence shows that at least the component connected to the source vertex of the traversed graph is acyclic.

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Another basic graph traversal algorithm is the O(V+E) Breadh-First Search (BFS) algorithm. As it name implies, BFS starts from a source vertex and uses a queue to order the visitation sequence as breadth as possible before going deeper. BFS also uses a Boolean array to distinguish visited and unvisited vertices.

In this visualization, we also show that starting from the same source vertex in an unweighted graph, BFS spanning tree of the graph equals to its SSSP spanning tree.

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We can use either the O(V+E) DFS or BFS to perform Topological Sort of a Directed Acyclic Graph (DAG).

The DFS version requires just one additional line compared to the normal DFS and is basically the post-order traversal of the graph.

The BFS version is based on the idea of vertices without incoming edge and is also called as Kahn's algorithm.

This topological sorting process is used internally in Dynamic Programming solution for SSSP on DAG.

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We can use either the O(V+E) DFS or BFS to check if a given graph is a Bipartite Graph by giving alternating color (orange versus blue in this visualization) between neighboring vertices and report 'non bipartite' if we ends up assigning same color to two adjacent vertices or 'bipartite' if it is possible to do such '2-coloring' process.

The DFS and BFS versions work similarly.

Bipartite Graphs have useful applications in (Bipartite) Graph Matching problem.

Note that Bipartite Graphs are usually only defined for undirected graphs so this visualization will convert directed input graphs into its undirected version automatically before continuing. This action is irreversible and you may have to redraw the directed input graph again for other purposes.

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We can modify the O(V+E) DFS algorithm into an algorithm to find Cut Vertices & Bridges of an Undirected Graph.

A Cut Vertex, or an Articulation Point, is a vertex of an undirected graph which removal disconnects the graph. Similarly, a bridge is an edge of an undirected graph which removal disconnects the graph.

Note that this algorithm for finding Cut Vertices & Bridges only works for undirected graphs so this visualization will convert directed input graphs into its undirected version automatically before continuing. This action is irreversible and you may have to redraw the directed input graph again for other purposes.

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We can modify the O(V+E) DFS algorithm into an algorithm to find Strongly Connected Components (SCCs) of a Directed Graph G.

An SCC of G a is defined as a subgraph S of G such that for any two vertices u and v in S, vertex u can reach vertex v directly or via a path, and vertex v can also reach vertex u back directly or via a path.

There are two known algorithms for finding SCCs of a Directed Graph: Kosaraju's and Tarjan's. Both of them are available in this visualization.

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We also have the 2-SAT Checker algorithm. Given a 2-Satisfiability (2-SAT) instance in the form of conjuction of clauses: (clause1) ^ (clause2) ^ ... ^ (clausen) and each clause is in form of disjunction of up to two variables (vara v varb), determine if we can assign True/False values to these variables so that the entire 2-SAT instance is evaluated to be true, i.e. satisfiable.

It turns out that each clause (a v b) can be turned into four vertices a, not a, b, and not b with two edges: (not a → b) and (not b → a). Thus we have a Directed Graph. If there is at least one variable and its negation inside an SCC of such graph, we know that it is impossible to satisfy the 2-SAT instance.

After such directed graph modeling, we can run an SCC finding algorithm (Kosaraju's or Tarjan's algorithm) to determine the satisfiability of the 2-SAT instance.

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As the action is being carried out, each step will be described in the status panel.

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You can also follow the pseudocode highlights to trace the algorithm.

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Control the animation with the player controls! Keyboard shortcuts are:

Spacebar: play/pause/replay
Left/right arrows: step backward/step forward
-/+: decrease/increase speed
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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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