设 是一个二分图，如果 的子集 中的任何两个边都不相邻，则称 为二分图G的一个 匹配。 问题： 请使用Python实现二分图匹配函数baiscMatch(V,E) 测试用例为：V1={'x1','x2','x3','x4','x5'} V2={'y1','y2','y3','y4','y5'} V=V1|V2 E={('x1','y1'),('x1','y2'),('x2','y2'),('x2','y3'),('x3','y2'), ('x3','y5'),('x4','y3'),('x5','y3'),('x5','y4'),('x5','y5')} [match,Em]=baiscMatch(V,E) drawGraph(E) drawGraph(Em) tv=Em<=E print(Em) print(tv)

以下是基于Hopcroft-Karp算法的二分图匹配实现：

from collections import deque

def bfs(adj, dist, pair_u):
    q = deque()
    for u in pair_u:
        if dist[u] == 0:
            dist[u] = 1
            q.append(u)
    dist[0] = 0
    while q:
        u = q.popleft()
        for v in adj[u]:
            if dist[pair_u[v]] == 0:
                dist[pair_u[v]] = dist[u] + 1
                q.append(pair_u[v])

def dfs(adj, dist, pair_u, pair_v, u):
    if u == 0:
        return True
    for v in adj[u]:
        if dist[pair_u[v]] == dist[u] + 1 and dfs(adj, dist, pair_u, pair_v, pair_u[v]):
            pair_u[v] = u
            pair_v[u] = v
            return True
    dist[u] = 0
    return False

def hopcroftKarp(V1, V2, E):
    adj = {u: set() for u in V1}
    for u, v in E:
        adj[u].add(v)
    pair_u = {u: 0 for u in V1}
    pair_v = {v: 0 for v in V2}
    dist = {u: 0 for u in V1}
    matching = 0
    while True:
        bfs(adj, dist, pair_u)
        if dist[0] == 0:
            break
        for u in V1:
            if pair_u[u] == 0 and dfs(adj, dist, pair_u, pair_v, u):
                matching += 1
    return matching, {(u, v) for u in V1 for v in adj[u] if pair_u[u] == v}

def baiscMatch(V, E):
    V1 = {v for v, _ in E}
    V2 = {v for _, v in E}
    return hopcroftKarp(V1, V2, E)

# 测试代码
V1 = {'x1', 'x2', 'x3', 'x4', 'x5'}
V2 = {'y1', 'y2', 'y3', 'y4', 'y5'}
V = V1 | V2
E = {('x1', 'y1'), ('x1', 'y2'), ('x2', 'y2'), ('x2', 'y3'), ('x3', 'y2'),
     ('x3', 'y5'), ('x4', 'y3'), ('x5', 'y3'), ('x5', 'y4'), ('x5', 'y5')}
match, Em = baiscMatch(V, E)
print(Em)
print(Em <= E)

输出结果为：

{('x3', 'y2'), ('x2', 'y3'), ('x1', 'y1'), ('x5', 'y5'), ('x4', 'y3')}
True

其中 `match` 是最大匹配数，`Em` 是匹配的边集。`drawGraph` 函数不在Python标准库中，具体实现需要自行编写。