Edmonds-Karp最大流算法

noHandFish

于 2024-12-19 14:31:39 发布

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文章标签：算法图论

本文链接：https://blog.youkuaiyun.com/weixin_58498239/article/details/144584926

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step1：初始化

创建剩余图 remain_graph使之和给出的graph相等，并设置最大流 max_flow 为 0。
初始化父节点数组 parent
广度优先搜索函数

remain_graph = capacity_matrix
n = len(capacity_matrix)
parent = [-1]*n
max_flow = 0

def bfs(remain_graph, s, t, parent):
    visited = set()
    queue = deque([s])
    visited.add(s)
    while queue:
        vertex = queue.popleft()
        for neighbour, capacity in enumerate(capacity_matrix[vertex]):
            if neighbour not in visited and capacity > 0:
                visited.add(neighbour)
                queue.append(neighbour)
                parent[neighbour] = vertex
                if neighbour == t:
                    return True
    return False

step2：使用BFS找到增广路径，计算路径流量（从汇点t回溯到s，找最小容量）

path_flow = float('inf')
start = t
while start != s:
        path_flow = min(path_flow, remain_graph[parent[start]][start])
        start = parent[start]

step3：更新残差网络，将路径流量加入最大流。

while v != s:
    u = parent[v]
    remain_graph[u][v] -= path_flow
    remain_graph[v][u] += path_flow
    v = u
max_flow += path_flow

step4：重复步骤2,3直至没有增广路径为止

完整算法代码如下：

def edmonds_karp(capacity_matrix, s, t):
    remain_graph = capacity_matrix
    n = len(capacity_matrix)
    parent = [-1]*n
    max_flow = 0
    while bfs(remain_graph, s, t, parent):
        path_flow = float('inf')
        start = t
        while start != s:
            path_flow = min(path_flow, remain_graph[parent[start]][start])
            start = parent[start]
        v = t
        while v != s:
            u = parent[v]
            remain_graph[u][v] -= path_flow
            remain_graph[v][u] += path_flow
            v = u
        max_flow += path_flow
    return max_flow