Bellman Ford

from __future__ import annotations


def print_distance(distance: list[float], src):
    print(f"Vertex\tShortest Distance from vertex {src}")
    for i, d in enumerate(distance):
        print(f"{i}\t\t{d}")


def check_negative_cycle(
    graph: list[dict[str, int]], distance: list[float], edge_count: int
):
    for j in range(edge_count):
        u, v, w = (graph[j][k] for k in ["src", "dst", "weight"])
        if distance[u] != float("inf") and distance[u] + w < distance[v]:
            return True
    return False


def bellman_ford(
    graph: list[dict[str, int]], vertex_count: int, edge_count: int, src: int
) -> list[float]:
    """
    Returns shortest paths from a vertex src to all
    other vertices.
    >>> edges = [(2, 1, -10), (3, 2, 3), (0, 3, 5), (0, 1, 4)]
    >>> g = [{"src": s, "dst": d, "weight": w} for s, d, w in edges]
    >>> bellman_ford(g, 4, 4, 0)
    [0.0, -2.0, 8.0, 5.0]
    >>> g = [{"src": s, "dst": d, "weight": w} for s, d, w in edges + [(1, 3, 5)]]
    >>> bellman_ford(g, 4, 5, 0)
    Traceback (most recent call last):
     ...
    Exception: Negative cycle found
    """
    distance = [float("inf")] * vertex_count
    distance[src] = 0.0

    for _ in range(vertex_count - 1):
        for j in range(edge_count):
            u, v, w = (graph[j][k] for k in ["src", "dst", "weight"])

            if distance[u] != float("inf") and distance[u] + w < distance[v]:
                distance[v] = distance[u] + w

    negative_cycle_exists = check_negative_cycle(graph, distance, edge_count)
    if negative_cycle_exists:
        raise Exception("Negative cycle found")

    return distance


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    V = int(input("Enter number of vertices: ").strip())
    E = int(input("Enter number of edges: ").strip())

    graph: list[dict[str, int]] = [{} for _ in range(E)]

    for i in range(E):
        print("Edge ", i + 1)
        src, dest, weight = (
            int(x)
            for x in input("Enter source, destination, weight: ").strip().split(" ")
        )
        graph[i] = {"src": src, "dst": dest, "weight": weight}

    source = int(input("\nEnter shortest path source:").strip())
    shortest_distance = bellman_ford(graph, V, E, source)
    print_distance(shortest_distance, 0)

About this Algorithm

Problem Statement

Given a weighted directed graph G(V,E) and a source vertex s ∈ V, determine for each vertex v ∈ V the shortest path between s and v.

Approach

• Initialize the distance from the source to all vertices as infinite.
• Initialize the distance to itself as 0.
• Create an array dist[] of size |V| with all values as infinite except dist[s].
• Repeat the following |V| – 1 times. Where |V| is number of vertices.
• Create another loop to go through each edge (u, v) in E and do the following:
  1. dist[v] = minimum(dist[v], dist[u] + weight of edge).
• Lastly iterate through all edges on last time to make sure there are no negatively weighted cycles.

Time Complexity

O(VE)

Space Complexity

O(V^2)

Founder’s Name

• Richard Bellman & Lester Ford, Jr.