A typescript implementation of the algorithm to find the maximum matching of the bipartite graph.

The following definition is quoted from Wikipedia (https://en.wikipedia.org/wiki/Matching_(graph_theory)):

A maximal matching is a matching $M$ of a graph $G$ that is not a subset of any other matching. A matching $M$ of a graph $G$ is maximal if every edge in $G$ has a non-empty intersection with at least one edge in $M$.

A maximum matching (also known as maximum-cardinality matching) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number $\displaystyle \nu (G)$ of a graph $G$ is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum matching. The following figure shows examples of maximum matchings in the same three graphs.

Install

• npm

  npm install --save @algorithm.ts/bipartite-graph-matching

• yarn

  yarn add @algorithm.ts/bipartite-graph-matching

• deno

  import { createBipartiteGraphMatching } from 'https://raw.githubusercontent.com/guanghechen/algorithm.ts/main/packages/bipartite-graph-matching/src/index.ts'

Usage

• Simple

  import { createBipartiteGraphMatching } from '@algorithm.ts/bipartite-graph-matching'
  
  const matching = createBipartiteGraphMatching()
  matching.init(4)
  matching.maxMatch() // => 0
  
  matching.addEdge(0, 1)
  matching.addEdge(0, 2)
  matching.addEdge(0, 3)
  matching.maxMatch() // => 1
  
  matching.addEdge(2, 3)
  matching.maxMatch() // => 2

Example

• A solution for leetcode "Maximum Students Taking Exam" (https://leetcode.com/problems/maximum-students-taking-exam/):

  import { createBipartiteGraphMatching } from '@algorithm.ts/bipartite-graph-matching'
  
  export function maxStudents(seats: string[][]): number {
    const R: number = seats.length
    if (R <= 0) return 0
  
    const C: number = seats[0].length
    if (C <= 0) return 0
  
    let total = 0
    const seatCodes: number[][] = new Array(R)
    for (let r = 0; r < R; ++r) seatCodes[r] = new Array(C).fill(-1)
  
    for (let r = 0; r < R; ++r) {
      for (let c = 0; c < C; ++c) {
        if (seats[r][c] === '.') {
          seatCodes[r][c] = total
          total += 1
        }
      }
    }
  
    if (total <= 0) return 0
    if (total === 1) return 1
  
    const matching = createBipartiteGraphMatching()
    matching.init(total)
    for (let r = 0; r < R; ++r) {
      for (let c = 0; c < C; ++c) {
        const u: number = seatCodes[r][c]
        if (u > -1) {
          if (r > 0) {
            // Check upper left
            if (c > 0 && seatCodes[r - 1][c - 1] > -1) {
              matching.addEdge(u, seatCodes[r - 1][c - 1])
            }
  
            // Check upper right
            if (c + 1 < C && seatCodes[r - 1][c + 1] > -1) {
              matching.addEdge(u, seatCodes[r - 1][c + 1])
            }
          }
  
          // Check left
          if (c > 0 && seatCodes[r][c - 1] > -1) {
            matching.addEdge(u, seatCodes[r][c - 1])
          }
        }
      }
    }
  
    const totalPaired: number = matching.maxMatch()
    return total - totalPaired
  }

Related

• 《算法竞赛入门经典（第2版）》（刘汝佳）： P347-P348 二分图最大匹配
• 二分图 | 光和尘
• Bipartite graph | Wikipedia
• Matching (graph theory) | Wikipedia