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In the Single-Source Shortest Paths (SSSP) problem, we aim to find the shortest paths weights (and the actual paths) from a particular single-source vertex to all other vertices in a directed weighted graph (if such paths exist).


The SSSP problem is a(nother) very well-known Computer Science (CS) problem that every CS students worldwide need to be aware of and hopefully master.


The SSSP problem has several different efficient (polynomial) algorithms (e.g. Bellman Ford, BFS, DFS, Dijkstra — 2 versions, and/or Dynamic Programming) that can be used depending on the nature of the input directed weighted graph, i.e. weighted/unweighted, with/without (negative weight) cycle, or structurally special (a tree/a DAG).


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SSSP is one of the most frequent graph problem encountered in real-life. Every time we want to move from one place (usually our current location) to another (our destination), we will try to pick a short — if not the shortest — path.


SSSP algorithm(s) is embedded inside various map software like Google Maps and in various Global Positioning System (GPS) tool.


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Input 1: A directed weighted graph G(V, E), not necessarily connected, where V/vertices can be used to describe intersections, junctions, houses, landmarks, etc and E/edges can be used to describe streets, roads, avenues with proper direction and weight/cost.


Input 2: As the name implies, the SSSP problem has another input: A source vertex sV.


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The objective of the SSSP problem is to find the shortest path weight from s to each vertex uV, denoted as δ(s, u) (δ is pronounced as 'delta') and also the actual shortest path from s to u.


The path weight of a path p is simply the summation of edge weights along that path.


The weight of the shortest path from s to s is trivial: 0.
The weight of the shortest path from s to any unreachable vertex is also trivial: +∞.


PS: The weight of the shortest path from s to v where (s, v) ∈ E does not necessarily the weight of w(s, v). See the next few slides to realise this.

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The outputs of all six (6) SSSP algorithms for the SSSP problem discussed in this visualization are these two arrays/Vectors:

  1. An array/Vector D of size V (D stands for 'distance')
    Initially, D[u] = 0 if u = s; otherwise D[u] = +∞ (a large number, e.g. 109)
    D[u] decreases as we find better (shorter) paths
    D[u]δ(s, u) throughout the execution of SSSP algorithm
    D[u] = δ(s, u) at the end of SSSP algorithm
  2. An array/Vector p of size V (p stands for 'parent'/'predecessor/previous')
    p[u] = the predecessor on best path from source s to u
    p[u] = NULL (not defined, we can use a value like -1 for this)
    This array/Vector p describes the resulting SSSP spanning tree
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Initially, D[u] = +∞ (practically, a large value like 109) ∀uV\{s}, but D[s] = D[0] = 0.
Initially, p[u] = -1 (to say 'no predecessor') ∀uV.


Now click Dijkstra(0) — don't worry about the details as they will be explained later — and wait until it is over (approximately 10s on this small graph).


At the end of that SSSP algorithm, D[s] = D[0] = 0 (unchanged) and D[u] = δ(s, u)uV
e.g. D[2] = 6, D[4] = 7 (these values are stored as red text under each vertex).
At the end of that SSSP algorithm, p[s] = p[0] = -1 (the source has no predecessor), but p[v] = the origin of the orange edges for the rest, e.g. p[2] = 0, p[4] = 2.


Thus, if we are at s = 0 and want to go to vertex 4, we will use shortest path is 0 → 2 → 4 with path weight 7.

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Some graphs contain negative weight edge(s) (not necessarily cyclic) and/or negative weight cycle(s). For example (fictional): Suppose you can travel forward in time (normal, edges with positive weight) or back in time by passing through time tunnel (special wormhole edges with negative weight), as the example shown above.


On that graph, the shortest paths from the source vertex s = 0 to vertices {1, 2, 3} are all ill-defined. For example 1 → 2 → 1 is a negative weight cycle as it has negative total path (cycle) weight of 15-42 = -27. Thus we can cycle around that negative weight cycle 0 → 1 → 2 → 1 → 2 → ... forever to get overall ill-defined shortest path weight of -∞.


However, notice that the shortest path from the source vertex s = 0 to vertex 4 is ok with δ(0, 4) = -99. So the presence of negative weight edge(s) is not the main issue. The main issue is the presence of negative weight cycle(s) reachable from source vertex s.

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The main operation for all SSSP algorithms discussed in this visualization is the relax(u, v, w(u, v)) operation with the following pseudo-code:

relax(u, v, w_u_v)
if D[v] > D[u]+w_u_v // if shortest path can be shortened
D[v] = D[u]+w_u_v // we 'relax' this edge
p[v] = u // remember/update the predecessor
// update some other data structure(s) as necessary

For example, see relax(1,2,4) operation on the figure below: relax operation example.

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There are two different sources for specifying an input graph:

  1. Draw Graph: You can draw any directed weighted graph as the input graph.
  2. Example Graphs: You can select from the list of our selected example graphs to get you started. These example graphs have different characteristics.
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In this visualization, we will discuss 6 (SIX) SSSP algorithms.


We will start with the O(V×E) Bellman Ford's algorithm first as it is the most versatile (but also the slowest) SSSP algorithm. We will then discuss 5 (FIVE) other algorithms (including two variants of Dijkstra's algorithm) that solve special-cases of SSSP problems in a much faster manner.

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The general purpose Bellman Ford's algorithm can solve all kinds of valid SSSP problem variants (expect one — the one that is ill-defined anyway, to be discussed soon), albeit with a rather slow O(V×E) running time. It also has an extremely simple pseudo-code:

for i = 1 to |V|-1 // O(V) here, so O(V×E×1) = O(V×E)
for each edge(u, v) ∈ E // O(E) here, e.g. by using an Edge List
relax(u, v, w(u, v)) // O(1) here

Without further ado, let's see a preview of how it works on the example graph above by clicking BellmanFord(0) (≈30s, and for now, please ignore the additional loop at the bottom of the pseudo-code).

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Bellman Ford's algorithm can be made to run slightly faster on normal input graph, from the worst case of O(V×E) to just O(k×E) where k is the number of iterations (of Bellman Ford's outer loop) used.


Discussion: How to do this? Is the speed-up significant?

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To convince the worldwide audience that Bellman Ford's algorithm works, let's temporarily move from visualization mode to proof mode for a few slides.


Theorem 1: If G = (V, E) contains no negative weight cycle, then the shortest path p from source vertex s to a vertex v must be a simple path.


Recall: A simple path is a path p = {v0, v1, v2, ..., vk}, (vi, vi+1) ∈ E, ∀ 0 ≤ i ≤ (k-1) and there is no repeated vertex along this path.

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Theorem 2: If G = (V, E) contains no negative weight cycle, then after Bellman Ford's algorithm terminates, we will have D[v] = δ(s, u), ∀ uV.


For this, we will use Proof by Induction and here are the starting points:


Consider the shortest path p from source vertex s to vertex vi where vi is defined as a vertex which the actual shortest path to reach it requires i hops (edges) from source vertex s. Recall from Theorem 1 that p will be simple path as we have the same assumption of no negative weight cycle.

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Try running BellmanFord(0) on the 'Bellman Ford's Killer' example above. There are V = 7 vertices and E = 6 edges but the edge list E is configured to be at its worst possible order. Notice that after (V-1)×E = (7-1)*6 = 36 operations (~40s, be patient), Bellman Ford's will terminate with the correct answer and there is no way we can terminate Bellman Ford's algorithm earlier.

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The only input graph that Bellman Ford's algorithm has issue is the input graph with negative weight cycle reachable from the source vertex s.


However, Bellman Ford's can be used to detect if the input graph contains at least one negative weight cycle reachable from the source vertex s by using the corollary of Theorem 2: If at least one value D[v] fails to converge after |V|-1 passes, then there exists a negative-weight cycle reachable from the source vertex s.


Now run BellmanFord(0) on the example graph that contains negative edges and negative weight cycles and concentrate on the loop at the bottom of the pseudo-code.

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Sometimes, the actual problem that we face is not the general form of the original problem. Therefore in this e-Lecture, we want to highlight five (5) special cases involving the SSSP problem. When we encounter any one of them, we can be solve it with different and (much) faster algorithm than the generic O(V×E) Bellman Ford's algorithm. They are:

  1. On Unweighted Graphs: O(V+E) BFS,
  2. On Graphs without negative weight: O((V+E) log V) Dijkstra's algorithm,
  3. On Graphs without negative weight cycle: O((V+E) log V) Modified Dijkstra's,
  4. On Tree: O(V+E) DFS/BFS,
  5. On Directed Acyclic Graphs (DAG): O(V+E) Dynamic Programming (DP)
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The O(V+E) Breadth-First Search (BFS) algorithm can solve special case of SSSP problem when the input graph is unweighted (all edges have unit weight 1, try BFS(5) on example: 'CP3 4.3' above) or positive constant weighted (all edges have the same constant weight, e.g. you can change all edge weights of the example graph above with any positive constant weight of your choice).

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When the graph is unweighted — this appears quite frequently in real life — the SSSP problem can be viewed as a problem of finding the least number of edges traversed from the source vertex s to other vertices.


The BFS spanning tree from source vertex s produced by the fast O(V+E) — notice the + sign — precisely fit the requirement.


Compared with the O(V×E) of Bellman Ford's — notice the × sign — it is a no-brainer to use BFS for this special case of SSSP problem.

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Compared to the standard BFS in Graph Traversal module, we need to perform simple modifications to make BFS able to solve the unweighted version of the SSSP problem:

  1. First, we change Boolean array visited to integer array D.
  2. At the start of BFS, instead of setting visited[u] = false, we set D[u] = 1e9 (a large number to symbolise +∞ or even -1 to symbolise 'unvisited' state, but we cannot use 0 as D[0] = 0) ∀uV\{s}; Then we set D[s] = 0
  3. We change the BFS main loop from
    if (visited[v] = 0) { visited[v] = 1 ... } // v is unvisited
    to
    if (D[v] = 1e9) { D[v] = D[u]+1 ... } // v is 1 step away from u
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However, BFS will very likely produce wrong answer when run on weighted graphs as BFS is not actually designed for to solve the weighted version of SSSP problem. There may be a case that taking a path with more number of edges used produces lower total overall path weight than taking a path with minimum number of edges used — which is the output of BFS algorithm.


In this visualization, we will allow you to run BFS even on 'wrong' input graph for pedagogical purpose, but we will display a warning message at the end of the algorithm. For example, try BFS(0) on the general graph above and you will see that vertices {3,4} will have wrong D[3] and D[4] values (and also p[3] and p[4] values).


We will soon see Dijkstra's algorithm (2 implementation variants) for solving certain weighted SSSP problems in a faster way than the general Bellman Ford's algorithm.

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The O((V+E) log V) Dijkstra's algorithm is the most frequently used SSSP algorithm for typical input: Directed weighted graph that has no negative weight edge at all, formally: ∀ edge(u, v) ∈ E, w(u, v) ≥ 0. Such weighted graph is very common in real life as travelling from one place to another always use positive time unit(s). Try Dijkstra(0) on one of the Example Graphs: CP3 4.17 shown above.

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Dijkstra's algorithm maintains a set S (Solved) of vertices whose final shortest path weights have been determined, initially S = {s}, the source vertex s only.


Then, it repeatedly selects vertex u in {V\S} with the minimum shortest path estimate, adds u to S, and relaxes all outgoing edges of u.


This entails the use of a Priority Queue as the shortest path estimates keep changing as more edges are processed. The choice of relaxing edges emanating from vertex with the minimum shortest path estimate first is greedy, i.e. use the "best so far", but we will see later that it can be proven that it will eventually ends up with an optimal result — if the graph has no negative weight edge.

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In Dijkstra's algorithm, each vertex will only be extracted from the Priority Queue (PQ) once. As there are V vertices, we will do this maximum O(V) times.


ExtractMin() operation runs in O(log V) whether the PQ is implemented using a Binary Min Heap or using a balanced BST like AVL Tree.


Therefore this part is O(V log V).

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Every time a vertex is processed, we relax its neighbors. In total, E edges are processed.


If by relaxing edge(u, v), we have to decrease D[v], we call the O(log V) DecreaseKey() operation in Binary Min Heap (harder to implement as C++ STL priority_queue/Java PriorityQueue does not support this operation efficiently yet) or simply delete the old entry and then re-insert a new entry in balanced BST like AVL Tree (which also runs in O(log V), but this is much easier to implement, just use C++ STL set/Java TreeSet).


Therefore, this part is O(E log V).


Thus in overall, Dijkstra's algorithm runs in O(V log V + E log V) = O((V+E) log V) time, which is much faster than the O(V×E) Bellman Ford's algorithm.

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When the input graph contains at least one negative weight edge — not necessarily negative weight cycle — Dijkstra's algorithm can produce wrong answer.


Try Dijkstra(0) on one of the Example Graphs: CP3 4.18.


At the end of the execution of Dijkstra's algorithm, vertex 4 has wrong D[4] value as the algorithm started 'wrongly' thinking that subpath 0 → 1 → 3 is the better subpath of weight 1+2 = 3, thus making D[4] = 6 after calling relax(3,4,3). However, the presence of negative weight -10 at edge 2 → 3 makes the other subpath 0 → 2 → 3 eventually the better subpath of weight 10-10 = 0 although it started worse with path weight 10 after the first edge 0 → 2. This better D[3] = 0 is never propagated further due to the greedy nature of Dijkstra's algorithm, hence D[4] is wrong.

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Dijkstra's algorithm can also be implemented differently. The O((V+E) log V) Modified Dijkstra's algorithm can be used for directed weighted graphs that may have negative weight edges but no negative weight cycle.


Such input graph appears in some practical cases, e.g. travelling using an electric car that has battery and our objective is to find a path from source vertex s to another vertex that minimizes overall battery usage. As usual, during acceleration (or driving on flat/uphill road), the electric car uses (positive) energy from the battery. However, during braking (or driving on downhill road), the electric car recharges (or use negative) energy to the battery. There is no negative weight cycle due to kinetic energy loss.


For example, try ModifiedDijkstra(0) on one of the Example Graphs: CP3 4.18 that has troubled the original version of Dijkstra's algorithm (see previous slide).

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When the input graph contains at least one negative weight edge but no negative weight cycle — the modified Dijkstra's algorithm produces correct answer.


Try ModifiedDijkstra(0) on one of the Example Graphs: CP3 4.18 that causes problem for Dijkstra(0).


At the end of the execution of ModifiedDijkstra's algorithm, vertex 4 has correct D[4] value as although the modified Dijkstra's algorithm also started 'wrongly' thinking that subpath 0 → 1 → 3 is the better subpath of weight 1+2 = 3, thus making D[4] = 6 after calling relax(3,4,3), the modified Dijkstra's algorithm continues propagating D[3] = 0 after it founds out that the other subpath 0 → 2 → 3 is eventually the better subpath of weight 10-10 = 0. Hence D[4] is eventually correct again. However, this is at the expense of potentially running (much more) operations than O((V+E) log V).

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Unfortunately, running ModifiedDijkstra(0) on the graph with negative weight cycle as shown on one of the Example Graphs: CP3 4.17 above will cause an endless loop (the animation is very long but we limit the number of loop to be 100 edges processed so your web browser will not hang).

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Try ModifiedDijkstra(0) on the extreme corner case above that is very hard to derive without proper understanding of this algorithm and was part of Asia Pacific Informatics Olympiad (APIO) 2013 task set by Steven.


The ModifiedDijkstra's algorithm will terminate with correct answer, but only after running exponential number of operations (each carefully constructed triangle raises the number of required operations by another power of two). Thus we cannot prematurely terminate ModifiedDijkstra's in this worst case input situation.


However, such extreme corner case is rare and thus in practice, Modified Dijkstra's algorithm can be used on directed graphs that have some negative weighted edges as long as the graph has no negative weight cycle reachable from the source vertex s.

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The O(V) Depth-First Search (DFS) algorithm can solve special case of SSSP problem, i.e. when the input graph is a (weighted) Tree.


In a Tree, there is only one unique and acylic path that connects two distinct vertices. Thus the unique path that connects the source vertex s to any another vertex uV is actually also the shortest path. For example, try DFS(0) on the Tree above.


Notice that for a (weighted) Tree, we can also use BFS. For example, try BFS(0) on the same Tree above.


Discussion: Why DFS (and also BFS) runs in O(V) instead of O(V+E) if the input is a (weighted) Tree?

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DFS will very likely produce wrong answer when run on any other graph that is not a Tree. We will display a warning message for such cases although we do not prevent you from trying this feature for pedagogical purpose.


For example, try DFS(0) on the general graph above and you will see that vertex {4} will have wrong D[4] value (and also p[4] value) as DFS(0) goes deep 0 → 1 → 3 → 4 first, backtrack all the way to vertex 0 and eventually visit 0 → 2 but edge 2 → 4 cannot be processed as vertex 4 has been visited by DFS earlier.

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The O(V+E) Dynamic Programming algorithm can solve special case of SSSP problem, i.e. when the input graph is a Directed Acyclic Graph (DAG) thus we can find at least one topological order of the DAG and process the edge relaxation according to this topological order.


For example, try DP(0) on the example DAG above. First, it computes one (there are other) possible topological order using either the O(V+E) DFS or the BFS/Kahn's algorithm outlined in Graph Traversal module. For example, assume one topological order is {0,2,1,3,4,5}. Then, it relaxes the outgoing edges of vertices listed in that topological order. After just one O(V+E) pass, we will have correct D[u] values ∀uV.

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On the Modified Dijkstra's killer example shown above, DP(0) works fast as the graph is actually a DAG, albeit having negative weight edge. As the graph is a DAG, there will not be any negative weight cycle to worry about.


However, DP will not work for any non DAG as non DAG contains at least one cycle and thus no topological order can be found within that cycle.

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DP algorithm for solving SSSP on DAG is also called one-pass Bellman Ford's algorithm as it replaces the outermost V-1 loop (we do not know the correct order so we just repeat until the maximum possible) with just one topological order pass (we know that this is (one of) the correct order(s) of this DAG).


Compare DP(0) (relax E edges just once — according to topological order of its vertices) versus BellmanFord(0) (relax E edges in random order, V-1 times) on the same example DAG above.

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We have lots of other stuffs on top of this basic explanation of SSSP algorithms for SSSP problems.


To get BellmanFordDemo.java/cpp, DijkstraDemo.java/cpp, visit Competitive Programming textbook website and get ch4.zip.

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For a few more interesting questions about this SSSP problem and its various algorithms, please practice on SSSP training module (no login is required, but short and of medium difficulty setting only).


However, for registered users, you should login and then go to the Main Training Page to officially clear this module (after clearing the other pre-requisites modules) and such achievement will be recorded in your user account.

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We also have a few programming problems that somewhat requires the usage of the correct SSSP algorithm: Kattis - hidingplaces and Kattis - shortestpath1.


Try to solve them and then try the many more interesting twists/variants of this interesting SSSP problem.


Advertisement: Buy Competitive Programming textbook to read more on this interesting problem.

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

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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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Draw Graph

Example Graphs

Bellman Ford's(s)

BFS Algorithm(s)

Dijkstra's Algorithm(s)

DFS Algorithm(s)

Dynamic Programming(s)

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CP3 4.3 U/U

CP3 4.4 D/U

CP3 4.17 D/W

CP3 4.18 -ve weight

CP3 4.19 -ve cycle

CP3 4.40 Tree

Bellman Ford's Killer

Dijkstra's Killer

DAG

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Original

Modified

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About Team Terms of use

About

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.

Team

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-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

List of translators who have contributed ≥100 translations can be found at statistics page.

Acknowledgements
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.