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__.

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

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:

**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),**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).**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:

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

PS1: Although possible, we will likely not use O(**V ^{3}**)

**Edmonds' Matching Algorithm**if the input is guaranteed to be a

**Bipartite Graph**.

PS2: Although possible, we will also likely not use O(**V ^{3}**)

**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(**V ^{2} + VE**) = O(

**VE**) implementation of that lemma (on Bipartite Graph): Find and then eliminate augmenting paths in Bipartite Graph

**G**. Click 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 **K _{N/2,N/2}**, i.e.,

**V = N/2+N/2 = N**and

**E = N/2×N/2 = N**, then

^{2}/4 ≈ N^{2}the Augmenting Path Algorithm discussed earlier will run in O(

**VE**) = O(

**N×N**) = O(

^{2}**N**).

^{3}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 , which is an almost complete **K _{5,5}** 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 **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

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 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:

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

We have not added visualizations for weighted variant of **MCBM** and **MCM** problems (future work).

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.

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Modeling

Example Graphs

Augmenting Path