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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 desirable but rarely possible result is Perfect Matching where all |V| vertices are matched (assuming |V| is even), 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.

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Graph Matching problems (and its variants) arise in various applications, e.g.,

  1. The underlying reason on why Social Development Network exists in Singapore
  2. Matching job openings (one disjoint set) to job applicants (the other disjoint set)
  3. The weighted version of #2 is called the Assignment problem
  4. Special-case of some NP-hard optimization problems
    (e.g., MVC, MIS, MPC on DAG, etc)
  5. Deterministic 2-opt Approximation Algorithm for MVC
  6. Sub-routine of Christofides's 1.5-approximation algorithm for TSP, etc...

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.

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In some applications, the weights of edges are not uniform (1 unit) but varies, and we may then want to take MCBM or MCM with minimum (or even maximum) total weight.


However, this visualization is currently limited to unweighted graphs only. Thus, we currently do not support Graph Matching problem variants involving weighted graphs...


The plan is to add Hungarian (Kuhn-Munkres) algorithm for weighted MCBM problem by April 2024, stay tuned...


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.

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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 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.

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You can view the visualisation here!


For Bipartite Graph visualization, we will mostly 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. When you draw your input bipartite graph, you can choose to re-layout your bipartite graph into this easier-to-visualize form. However, you do not have to visualize Bipartite Graph in this form, e.g., you can click Grid Graph to load an example grid graph and notice that vertices {0,1,2,3} can form set U and vertices {4,5,6,7,8} can form set V. There is no odd-length cycle in this grid graph.


For General Graph, we do not (and usually cannot) relayout the vertices into this Left Set and Right Set form.


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.

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There are three different sources for specifying an input graph:

  1. Edit Graph: You can draw any undirected unweighted graph as the input graph.
    However, due to the way we visualize our MCBM algorithms, we need to impose one additional graph drawing constraint that does not exist in the actual MCBM problems. That constraint is that vertices on the left set are numbered from [0, n), and vertices on the right set are numbered from [n, n+m). You do not have to visually draw them in left-right sets form, as shown in this Grid Graph example.
  2. Modeling: Several 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. Example Graphs: 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.
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This slide is a stub and will be expanded with the explanation of this problem and how to interpret the bipartite graph created.

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You can create any (small) bipartite graph with n/m vertices on the left set, respectively, and set the density of the edges, with 100% being a complete bipartite graph Kn,m and 0% being a bipartite graph with no edge.

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

  1. By reducing MCBM problem into a Max-Flow problem in polynomial time,
    we can actually use any Max Flow algorithm to solve MCBM.
  2. O(VE) Augmenting Path Algorithm (without greedy pre-processing),
  3. O(√(V)E) Dinic's or 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 (as it is much slower).

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 (again, as it is much slower).

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The MCBM problem can be modeled (or reduced into) as a Max Flow problem in polynomial time.


Go to Max Flow visualization page and see the flow graph modeling of MCBM problem (select Modeling → Bipartite Matching → all 1). Basically, create a super source vertex s that connects to all vertices in the left set and also create a super sink vertex t where all vertices in the right set connect to t. Keep all edges in the flow graph directed from source to sink and with unit weight 1.


If we use one of the earliest Max Flow algorithm, i.e., a simple Ford-Fulkerson algorithm, the time complexity will be tighter than O(mf × E) as all edge weights in the flow graph are unit weight so mf ≤ V, i.e., so O(V × E) overall.


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 — the analysis is omitted for now. This allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions.


Discussion: The edges in the flow graph must be directed. Why?
Then prove that this reduction is correct.

🕑

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.

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Actually, we can just stop here, i.e., when given any MCBM(-related) problem, we can simply reduce it into a Max-Flow problem and use (the fastest) Max Flow algorithm.


However, there is a far simpler Graph Matching algorithm that we will see in the next few slides. It is based on a crucial theorem and can be implemented as an easy variation of the standard Depth-First Search (DFS) algorithm.

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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 although augmenting path, if any, is much easier to find in a bipartite graph), alternates through unmatched (or free/'f'), matched (or 'm'), ..., unmatched ('f') edges in G, until it ends at another free vertex v. The pattern of any Augmenting Path will be fmf...fmf and is of odd length.


If we flip the edge status along that augmenting path, i.e., fmf...fmf into mfm...mfm, we will increase the number of edges in the matching set M by exactly 1 unit and eliminates this augmenting path.


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


Discussion: In class, prove the correctness of Berge's theorem!
In practice, we can just use it verbatim.

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The proof claims if and only if, thus it has two parts:
the forwards direction and the backwards direction.


The forwards proof is easier:
M∈G is maximum → there is no augmenting path in G w.r.t M.


The backwards proof is a bit harder:
M∈G is maximum ← there is no augmenting path in G w.r.t M.

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

Recall: Berge's theorem states:
A matching M in graph G is maximum iff there is no more augmenting path in G.


The Augmenting Path Algorithm (on Bipartite Graph) is a simple O(V*(V+E)) = O(V2 + VE) = O(VE) implementation (a modification of DFS) of that theorem: Find and then eliminate augmenting paths in Bipartite Graph G.


Click Augmenting Path Algorithm Demo to visualize this algorithm on a special test case called X̄ (X-bar).


Basically, this Augmenting Path Algorithm scans through all vertices on the left set (that were initially free vertices) one by one. Suppose L on the left set is a free vertex, this algorithm will recursively (via modification of DFS) go to a vertex R on the right set:

  1. If R is another free vertex, we have found one augmenting path (e.g., Augmenting Path 0-2 initially), and
  2. If R is already matched (this information is stored at match[R]), we immediately return to the left set and recurse (e.g, path 1-2-immediately return to 0-then 0-3, to find the second Augmenting Path 1-2-0-3)
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vi match, vis;           // global variables

int Aug(int L) { // notice similarities with DFS algorithm
if (vis[L]) return 0; // L visited, return 0
vis[L] = 1;
for (auto& R : AL[L])
if ((match[R] == -1) || Aug(match[R])) { // the key modification
match[R] = L; // flip status
return 1; // found 1 matching
}
return 0; // Augmenting Path is not found
}
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// 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) { // try all left vertices
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.

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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.


It feels bad, especially on the latter iterations...
So, should we avoid using this simple Augmenting Path algorithm?

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The key idea of Hopcroft-Karp (HK) Algorithm (invented in 1973) is identical to Dinic's Max Flow Algorithm, 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.


Since Hopcroft-Karp algorithm is essentially also Dinic's algorithm, we treat both as 'approximately equal'.

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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 eliminate 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 Plus 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.

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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: Example Graphs, Corner Case, Rand Greedy AP 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 Plus also allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions.

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So, when presented with an MCBM problem, which route should we take?

  1. Reduce the MCBM problem into Max-Flow and use Dinic's algorithm (essentially Hopcroft-Karp algorithm) and gets O(√(V)E) performance guarantee but with a much longer implementation?
  2. Use Augmenting Path algorithm with Randomized Greedy Processing with O(kE) performance with good empirical results and a much shorter implementation?

Discussion: Discuss these two routes!

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

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.

🕑

Unfortunately there is no weighted MCBM algorithm (e.g., Min-Cost-Max-Flow (mcmf) or Hungarian/Kuhn-Munkres) visualization in VisuAlgo yet. But the plan is to have this visualization eventually.


For this section, please refer to CP4 Book 2 Chapter 9.25 and 9.27.

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When Graph Matching is posed on general graphs (the MCM problem), it is (much) harder to find Augmenting Path. In fact, before Jack Edmonds published his famous paper titled "Paths, Trees, and Flowers" in 1965, this MCM problem was thought to be an (NP-)hard optimization problem.


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

  1. O(V^3) Edmonds' Matching algorithm (without greedy pre-processing),
  2. O(V^3) Edmonds' Matching algorithm (with greedy pre-processing),
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In General Graph (like the graph shown in the background that has |MCM| = 4), we may have Odd-Length cycle. Augmenting Path is not well defined in such a graph, hence we cannot easily implement Claude Berge's theorem 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 a Blossom. This situation is only possible if we have Odd-Length cycle, i.e., in a non-Bipartite Graph. For example, assume edge 1-2 has been matched in the graph shown in the background, then path 3-1=2-3 is a blossom.


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 as the current visualization of Edmonds' matching algorithm in VisuAlgo is still 'a bit too hard too understand' for beginners, try Edmonds' Matching. In a live class in NUS, these steps will be explained verbally.


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

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O(V^3) Edmonds' Matching Algorithm Plus

As with the Augmenting Path Algorithm Plus 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.

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We have not added the visualizations for weighted variant of MCBM and MCM problems. They are for future work. Among the two, weighted MCBM will likely be added earlier than the weighted MCM version.


One of the possible solution for weighted MCBM problem is the Hungarian algorithm. This algorithm also has another name: The Kuhn-Munkres algorithm. This algorithm relies on Berge's Augmenting Path Theorem too, but it uses the theorem slightly differently.

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To strengthen your understanding about these Graph Matching problem, its variations, and the multiple possible solutions, please try solving as many of these programming competition problems listed below:

  1. Standard MCBM (but need a fast algorithm): Kattis - flippingcards
  2. Greedy Bipartite Matching: Kattis - froshweek2
    (you do not need a specific MCBM algorithm for this,
    in fact, it will be too slow if you use any algorithm discussed here)
  3. Special case of an NP-hard optimization problem: Kattis - bilateral
  4. Rather straightforward weighted MCBM: Kattis - engaging
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To tackle those programming contest problems, you are allowed to use/modify our implementation code for Augmenting Path Algorithm (with Randomized Greedy Preprocessing): 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.

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编辑图表

Modeling

图示

增广路

>

Rook Attack

Generate Random Bipartite Graph, specify n, m, and edge density

Generate Random Weighted Bipartite Graph, specify n

K2,2

F-mod

Corner Case

Special Case

Performance Test

Matching with Capacity

waif (WA)

CP4 3.11a*

Theorem

Sample Weighted Bipartite

Sample Weighted Bipartite TUM

标准

带有随机贪心预处理

Hopcroft Karp

Edmonds Blossom

Edmonds Blossom + Greedy

Hungarian

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关于

VisuAlgo最初由副教授Steven Halim于2011年构思,旨在通过提供自学、互动式学习平台,帮助学生更深入地理解数据结构和算法。

VisuAlgo涵盖了Steven Halim博士与Felix Halim博士、Suhendry Effendy博士合著的书《竞技编程》中讨论的许多高级算法。即使过去十年,VisuAlgo仍然是可视化和动画化这些复杂算法的独家平台。

虽然VisuAlgo主要面向新加坡国立大学(NUS)的学生,包括各种数据结构和算法课程(例如CS1010/等价课程,CS2040/等价课程(包括IT5003),CS3230,CS3233和CS4234),但它也是全球好奇心的宝贵资源,促进在线学习。

最初,VisuAlgo并不适用于智能手机等小触摸屏,因为复杂的算法可视化需要大量的像素空间和点击拖动交互。为了获得最佳用户体验,建议使用最低分辨率为1366x768的屏幕。然而,自2022年4月以来,VisuAlgo的移动(精简)版本已经推出,使得在智能手机屏幕上使用VisuAlgo的部分功能成为可能。

VisuAlgo仍然在不断发展中,正在开发更复杂的可视化。目前,该平台拥有24个可视化模块。

VisuAlgo配备了内置的问题生成器和答案验证器,其“在线测验系统”使学生能够测试他们对基本数据结构和算法的理解。问题根据特定规则随机生成,并且学生提交答案后会自动得到评分。随着越来越多的计算机科学教师在全球范围内采用这种在线测验系统,它可以有效地消除许多大学标准计算机科学考试中手工基本数据结构和算法问题。通过给通过在线测验的学生分配一个小但非零的权重,计算机科学教师可以显著提高学生对这些基本概念的掌握程度,因为他们可以在参加在线测验之前立即验证几乎无限数量的练习题。每个VisuAlgo可视化模块现在都包含自己的在线测验组件。

VisuAlgo已经被翻译成三种主要语言:英语、中文和印尼语。此外,我们还用各种语言撰写了关于VisuAlgo的公开笔记,包括印尼语、韩语、越南语和泰语:

id, kr, vn, th.

团队

项目领导和顾问(2011年7月至今)
Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

本科生研究人员 1
CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

最后一年项目/ UROP学生 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

本科生研究人员 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

最后一年项目/ UROP学生 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

本科生研究人员 3
Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

最后一年项目/ UROP学生 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.

致谢
NUS教学与学习发展中心(CDTL)授予拨款以启动这个项目。在2023/24学年,Optiver的慷慨捐赠将被用来进一步开发 VisuAlgo。

使用条款

VisuAlgo慷慨地向全球计算机科学界提供免费服务。如果您喜欢VisuAlgo,我们恳请您向其他计算机科学学生和教师宣传它的存在。您可以通过社交媒体平台(如Facebook、YouTube、Instagram、TikTok、Twitter等)、课程网页、博客评论、电子邮件等方式分享VisuAlgo。

数据结构与算法(DSA)的学生和教师可以直接在课堂上使用本网站。如果您从本网站截取屏幕截图或视频,可以在其他地方使用,但请引用本网站的URL(https://visualgo.net)和/或下面的出版物列表作为参考。但请不要下载VisuAlgo的客户端文件并将其托管在您的网站上,因为这构成了抄袭行为。目前,我们不允许他人分叉此项目或创建VisuAlgo的变体。个人使用离线副本的客户端VisuAlgo是可以接受的。

请注意,VisuAlgo的在线测验组件具有重要的服务器端元素,保存服务器端脚本和数据库并不容易。目前,普通公众只能通过“培训模式”访问在线测验系统。“测试模式”提供了一个更受控制的环境,用于在新加坡国立大学的真实考试中使用随机生成的问题和自动验证。


出版物列表

这项工作曾在2012年国际大学生程序设计竞赛(波兰,华沙)的CLI研讨会上和2012年国际信息学奥林匹克竞赛(意大利,锡尔米奥内-蒙蒂基亚里)的IOI会议上展示过。您可以点击此链接阅读我们2012年关于该系统的论文(当时还没有称为VisuAlgo),以及此链接阅读2015年的简短更新(将VisuAlgo与之前的项目关联起来)。


错误报告或新功能请求

VisuAlgo并不是一个完成的项目。Steven Halim副教授仍在积极改进VisuAlgo。如果您在使用VisuAlgo时发现任何可视化页面/在线测验工具中的错误,或者您想要请求新功能,请联系Steven Halim副教授。他的联系方式是将他的名字连接起来,然后加上gmail dot com。

隐私政策

版本 1.2 (更新于2023年8月18日星期五)。

自2023年8月18日(星期五)起,我们不再使用 Google Analytics。因此,我们现在使用的所有 cookies 仅用于此网站的运营。即使是首次访问的用户,烦人的 cookie 同意弹窗现在也已关闭。

自2023年6月7日(星期五)起,由于 Optiver 的慷慨捐赠,全世界的任何人都可以自行创建一个 VisuAlgo 账户,以存储一些自定义设置(例如,布局模式,默认语言,播放速度等)。

此外,对于 NUS 学生,通过使用 VisuAlgo 账户(一个 NUS 官方电子邮件地址,课堂名册中的学生姓名,以及在服务器端加密的密码 - 不存储其他个人数据),您同意您的课程讲师跟踪您的电子讲义阅读和在线测验培训进度,这是顺利进行课程所必需的。您的 VisuAlgo 账户也将用于参加 NUS 官方的 VisuAlgo 在线测验,因此,将您的账户凭据传递给他人代您进行在线测验构成学术违规。课程结束后,您的用户账户将被清除,除非您选择保留您的账户(OPT-IN)。访问完整的 VisuAlgo 数据库(包含加密密码)的权限仅限于 Halim 教授本人。

对于全球其他已经给 Steven 写过信的 CS 讲师,需要一个 VisuAlgo 账户(您的(非 NUS)电子邮件地址,您可以使用任何显示名称,以及加密密码)来区分您的在线凭据与世界其他地方。您的账户将具有 CS 讲师特定的功能,即能够查看隐藏的幻灯片,这些幻灯片包含了在隐藏幻灯片之前的幻灯片中提出的问题的(有趣的)答案。您还可以访问 VisuAlgo 在线测验的 Hard 设置。您可以自由地使用这些材料来增强您的数据结构和算法课程。请注意,未来可能会有其他 CS 讲师特定的功能。

对于任何拥有 VisuAlgo 账户的人,如果您希望不再与 VisuAlgo 工具有关联,您可以自行删除您的账户。