# Which Data Structure Is Used for Bipartite Perfect Matching Problem?

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Larry Thompson

Which Data Structure Is Used for Bipartite Perfect Matching Problem?

When it comes to solving the bipartite perfect matching problem, choosing the right data structure is crucial. In this article, we will explore some of the commonly used data structures that are well-suited for this problem.

An adjacency matrix is a popular choice for representing a bipartite graph.

It provides a compact way to store the connections between vertices. In the context of bipartite matching, an adjacency matrix can be used to represent the edges between two sets of vertices.

To construct an adjacency matrix for a bipartite graph, we can create a 2D array with rows and columns corresponding to each vertex in both sets. The value at position (i, j) in the matrix represents whether there is an edge between vertex i in one set and vertex j in the other set.

Example:

```    A   B   C   D
X   1   0   1   0
Y   0   1   0   1
Z   1   0   0   1
```

Another commonly used data structure for bipartite matching is an adjacency list. Unlike an adjacency matrix, which uses a dense representation, an adjacency list provides a more memory-efficient way to store sparse graphs.

In an adjacency list, each vertex is associated with a list of its neighboring vertices. For bipartite matching, we can maintain separate lists for vertices in each set.

Example:

```X: [A, C]
Y: [B, D]
Z: [A, D]
```

## 3. Flow Network

In some cases, the bipartite perfect matching problem can be reduced to a maximum flow problem on a flow network. A flow network consists of a directed graph with capacities on its edges.

To solve the bipartite perfect matching problem using a flow network, we can add a source node connected to all vertices in one set and a sink node connected to all vertices in the other set. The capacities of these edges are set to 1.

Example:

```           (s)
/  |  \
/   |   \
A    B    C
\   |   /
\  |  /
(t)
```

## 4. Bipartite Matching Algorithm

Once we have chosen an appropriate data structure, we can apply an algorithm to find the bipartite perfect matching. One commonly used algorithm is the Hopcroft-Karp algorithm, which has a time complexity of O(sqrt(V) * E).

The Hopcroft-Karp algorithm starts with an empty matching and repeatedly augments it until no more augmenting paths can be found. It uses breadth-first search (BFS) to find augmenting paths efficiently.

### Summary

In conclusion, when solving the bipartite perfect matching problem, choosing the right data structure is essential for efficient and accurate results. The adjacency matrix and adjacency list are popular choices for representing bipartite graphs.

In some cases, reducing the problem to a maximum flow problem on a flow network can also be beneficial. Finally, applying algorithms like Hopcroft-Karp can help find the desired matching efficiently.