Python/graphs/johnson_graph.py

import heapq
import sys


# First implementation of johnson algorithm
# Steps followed to implement this algorithm is given in the below link:
# https://brilliant.org/wiki/johnsons-algorithm/
class JohnsonGraph:
    def __init__(self) -> None:
        """
        Initializes an empty graph with no edges.
        """
        self.edges: list[tuple[str, str, int]] = []
        self.graph: dict[str, list[tuple[str, int]]] = {}

    # add vertices for a graph
    def add_vertices(self, vertex: str) -> None:
        """
        Adds a vertex `u` to the graph with an empty adjacency list.
        """
        self.graph[vertex] = []

    # assign weights for each edges formed of the directed graph
    def add_edge(self, vertex_a: str, vertex_b: str, weight: int) -> None:
        """
        Adds a directed edge from vertex `u` to vertex `v` with weight `w`.
        """
        self.edges.append((vertex_a, vertex_b, weight))
        self.graph[vertex_a].append((vertex_b, weight))

    # perform a dijkstra algorithm on a directed graph
    def dijkstra(self, start: str) -> dict:
        """
        Computes the shortest path from vertex `s`
        to all other vertices using Dijkstra's algorithm.
        """
        distances = {vertex: sys.maxsize - 1 for vertex in self.graph}
        pq = [(0, start)]
        distances[start] = 0
        while pq:
            weight, vertex = heapq.heappop(pq)

            if weight > distances[vertex]:
                continue

            for node, node_weight in self.graph[vertex]:
                if distances[vertex] + node_weight < distances[node]:
                    distances[node] = distances[vertex] + node_weight
                    heapq.heappush(pq, (distances[node], node))
        return distances

    # carry out the bellman ford algorithm for a node and estimate its distance vector
    def bellman_ford(self, start: str) -> dict:
        """
        Computes the shortest path from vertex `s`
        to all other vertices using the Bellman-Ford algorithm.
        """
        distances = {vertex: sys.maxsize - 1 for vertex in self.graph}
        distances[start] = 0

        for vertex_a in self.graph:
            for vertex_a, vertex_b, weight in self.edges:
                if (
                    distances[vertex_a] != sys.maxsize - 1
                    and distances[vertex_a] + weight < distances[vertex_b]
                ):
                    distances[vertex_b] = distances[vertex_a] + weight

        return distances

    # perform the johnson algorithm to handle the negative weights that
    # could not be handled by either the dijkstra
    # or the bellman ford algorithm efficiently
    def johnson_algo(self) -> list[dict]:
        """
        Computes the shortest paths between
        all pairs of vertices using Johnson's algorithm.
        """
        self.add_vertices("#")
        for vertex in self.graph:
            if vertex != "#":
                self.add_edge("#", vertex, 0)

        hash_path = self.bellman_ford("#")

        for i in range(len(self.edges)):
            vertex_a, vertex_b, weight = self.edges[i]
            self.edges[i] = (
                vertex_a,
                vertex_b,
                weight + hash_path[vertex_a] - hash_path[vertex_b],
            )
            self.edges[i] = (vertex_a, 
                            vertex_b, 
                            weight + hash_path[vertex_a] - hash_path[vertex_b])

        self.graph.pop("#")
        self.edges = [
            (vertex1, vertex2, node_weight)
            for vertex1, vertex2, node_weight in self.edges
            if vertex1 != "#"
        ]
        filtered_edges = []
        for vertex1, vertex2, node_weight in self.edges:
            if vertex1 != "#":
                filtered_edges.append((vertex1, vertex2, node_weight))
        self.edges = filtered_edges

        for vertex in self.graph:
            self.graph[vertex] = [
                (vertex2, node_weight)
                for vertex1, vertex2, node_weight in self.edges
                if vertex1 == vertex
            ]

            filtered_neighbors = []
            for vertex1, vertex2, node_weight in self.edges:
                if vertex1 == vertex:
                    filtered_neighbors.append((vertex2, node_weight))
            self.graph[vertex] = filtered_neighbors
            
        distances = []
        for vertex1 in self.graph:
            new_dist = self.dijkstra(vertex1)
            for vertex2 in self.graph:
                if new_dist[vertex2] < sys.maxsize - 1:
                    new_dist[vertex2] += hash_path[vertex1] - hash_path[vertex2]
            distances.append(new_dist)
        return distances


g = JohnsonGraph()
# this a complete connected graph
g.add_vertices("A")
g.add_vertices("B")
g.add_vertices("C")
g.add_vertices("D")
g.add_vertices("E")

g.add_edge("A", "B", 1)
g.add_edge("A", "C", 3)
g.add_edge("B", "D", 4)
g.add_edge("D", "E", 2)
g.add_edge("E", "C", -2)


optimal_paths = g.johnson_algo()
print("Print all optimal paths of a graph using Johnson Algorithm")
for i, row in enumerate(optimal_paths):
    print(f"{i}: {row}")