Implementation Details

Here is the Edmonds-Karp implementation structure:

function edmondsKarp(n, source, sink, capacity):
    flow = 0

    while true:
        // BFS to find augmenting path
        parent = array of size n, all -1
        visited = array of size n, all false
        queue.push(source)
        visited[source] = true

        while queue is not empty and not visited[sink]:
            u = queue.pop()
            for v in adj[u]:
                if not visited[v] and capacity[u][v] > 0:
                    visited[v] = true
                    parent[v] = u
                    queue.push(v)

        if not visited[sink]:
            break

        // Find bottleneck
        bottleneck = infinity
        v = sink
        while v != source:
            u = parent[v]
            bottleneck = min(bottleneck, capacity[u][v])
            v = u

        // Update capacities
        v = sink
        while v != source:
            u = parent[v]
            capacity[u][v] = capacity[u][v] - bottleneck
            capacity[v][u] = capacity[v][u] + bottleneck
            v = u

        flow = flow + bottleneck

    return flow

Time: $O(VE^2)$. Space: $O(V + E)$.