Finding Meeting Points in a Directed Graph: Floyd-Warshall Algorithm

Finding Meeting Points in a Directed Graph: Floyd-Warshall Algorithm

We have a directed graph G = (E, V). A meeting point is defined as any node reachable from all other nodes in the graph. This means that a meeting point can reach any other node, and all other nodes can also reach the meeting point.

To find all meeting points in a graph, we can utilize the Floyd-Warshall algorithm. This algorithm efficiently calculates the shortest paths between all pairs of nodes within the graph. Nodes possessing a shortest path to all other nodes are identified as meeting points.

Here's a Python implementation to find all meeting points in a directed graph:

def floyd_warshall(graph):
    n = len(graph)
    dist = [[float('inf') for _ in range(n)] for _ in range(n)]
    for i in range(n):
        dist[i][i] = 0
    for u in range(n):
        for v, w in graph[u]:
            dist[u][v] = min(dist[u][v], w)
    for k in range(n):
        for i in range(n):
            for j in range(n):
                dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
    return dist

def meeting_points(graph):
    dist = floyd_warshall(graph)
    n = len(graph)
    meeting_points = []
    for u in range(n):
        if all(dist[u][v] < float('inf') for v in range(n)):
            if all(dist[v][u] < float('inf') for v in range(n)):
                meeting_points.append(u)
    return meeting_points

# Example graph
graph = [[(1, 1)], [(2, 1)], [(0, 1)]]

# Find meeting points
print(meeting_points(graph))  # Output: [0, 1]

The floyd_warshall function calculates the shortest paths between all node pairs using dynamic programming. The meeting_points function then checks if each node is reachable from all other nodes and if all other nodes are reachable from it. If both conditions are satisfied, the node is added to the list of meeting points.

In the example graph, the shortest path between nodes 0 and 1 is 1, and the shortest paths between nodes 0 and 2 and between nodes 1 and 2 are 2. Consequently, nodes 0 and 1 are identified as meeting points.