Johnson’s algorithm for All-pairs shortest paths | Implementation

Given a weighted Directed Graph where the weights may be negative, find the shortest path between every pair of vertices in the Graph using Johnson’s Algorithm.

The detailed explanation of Johnson’s algorithm has already been discussed in the previous post.

Refer: Johnson’s algorithm for All-pairs shortest paths.



This post focusses on the implementation of Johnson’s Algorithm.

Algorithm:

  1. Let the given graph be G. Add a new vertex s to the graph, add edges from new vertex to all vertices of G. Let the modified graph be G’.
  2. Run Bellman-Ford algorithm on G’ with s as source. Let the distances calculated by Bellman-Ford be h[0], h[1], .. h[V-1]. If we find a negative weight cycle, then return. Note that the negative weight cycle cannot be created by new vertex s as there is no edge to s. All edges are from s.
  3. Reweight the edges of original graph. For each edge (u, v), assign the new weight as “original weight + h[u] – h[v]”.
  4. Remove the added vertex s and run Dijkstra’s algorithm for every vertex.

Example:
Let us consider the following graph.

Johnson1

We add a source s and add edges from s to all vertices of the original graph. In the following diagram s is 4.

Johnson2

We calculate the shortest distances from 4 to all other vertices using Bellman-Ford algorithm. The shortest distances from 4 to 0, 1, 2 and 3 are 0, -5, -1 and 0 respectively, i.e., h[] = {0, -5, -1, 0}. Once we get these distances, we remove the source vertex 4 and reweight the edges using following formula. w(u, v) = w(u, v) + h[u] – h[v].

Johnson3

Since all weights are positive now, we can run Dijkstra’s shortest path algorithm for every vertex as source.

Below is the implementation of the above approach

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# Implementation of Johnson's algorithm in Python3
  
# Import function to initialize the dictionary
from collections import defaultdict
MAX_INT = float('Inf')
  
# Returns the vertex with minimum 
# distance from the source
def minDistance(dist, visited):
  
    (minimum, minVertex) = (MAX_INT, 0)
    for vertex in range(len(dist)):
        if minimum > dist[vertex] and visited[vertex] == False:
            (minimum, minVertex) = (dist[vertex], vertex)
  
    return minVertex
  
  
# Dijkstra Algorithm for Modified 
# Graph (removing negative weights)
def Dijkstra(graph, modifiedGraph, src):
  
    # Number of vertices in the graph
    num_vertices = len(graph)
  
    # Dictionary to check if given vertex is 
    # already included in the shortest path tree
    sptSet = defaultdict(lambda : False)
  
    # Shortest distance of all vertices from the source
    dist = [MAX_INT] * num_vertices
  
    dist[src] = 0
  
    for count in range(num_vertices):
  
        # The current vertex which is at min Distance 
        # from the source and not yet included in the 
        # shortest path tree
        curVertex = minDistance(dist, sptSet)
        sptSet[curVertex] = True
  
        for vertex in range(num_vertices):
            if ((sptSet[vertex] == False) and
                (dist[vertex] > (dist[curVertex] + 
                modifiedGraph[curVertex][vertex])) and
                (graph[curVertex][vertex] != 0)):
                  
                dist[vertex] = (dist[curVertex] +
                                modifiedGraph[curVertex][vertex]);
  
    # Print the Shortest distance from the source
    for vertex in range(num_vertices):
        print ('Vertex ' + str(vertex) + ': ' + str(dist[vertex]))
  
# Function to calculate shortest distances from source
# to all other vertices using Bellman-Ford algorithm
def BellmanFord(edges, graph, num_vertices):
  
    # Add a source s and calculate its min
    # distance from every other node
    dist = [MAX_INT] * (num_vertices + 1)
    dist[num_vertices] = 0
  
    for i in range(num_vertices):
        edges.append([num_vertices, i, 0])
  
    for i in range(num_vertices):
        for (src, des, weight) in edges:
            if((dist[src] != MAX_INT) and 
                    (dist[src] + weight < dist[des])):
                dist[des] = dist[src] + weight
  
    # Don't send the value for the source added
    return dist[0:num_vertices]
  
# Function to implement Johnson Algorithm
def JohnsonAlgorithm(graph):
  
    edges = []
  
    # Create a list of edges for Bellman-Ford Algorithm
    for i in range(len(graph)):
        for j in range(len(graph[i])):
  
            if graph[i][j] != 0:
                edges.append([i, j, graph[i][j]])
  
    # Weights used to modify the original weights
    modifyWeights = BellmanFord(edges, graph, len(graph))
  
    modifiedGraph = [[0 for x in range(len(graph))] for y in
                    range(len(graph))]
  
    # Modify the weights to get rid of negative weights
    for i in range(len(graph)):
        for j in range(len(graph[i])):
  
            if graph[i][j] != 0:
                modifiedGraph[i][j] = (graph[i][j] + 
                        modifyWeights[i] - modifyWeights[j]);
  
    print ('Modified Graph: ' + str(modifiedGraph))
  
    # Run Dijkstra for every vertex as source one by one
    for src in range(len(graph)):
        print ('\nShortest Distance with vertex ' +
                        str(src) + ' as the source:\n')
        Dijkstra(graph, modifiedGraph, src)
  
# Driver Code
graph = [[0, -5, 2, 3], 
         [0, 0, 4, 0], 
         [0, 0, 0, 1], 
         [0, 0, 0, 0]]
  
JohnsonAlgorithm(graph)

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Output:

Modified Graph: [[0, 0, 3, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]

Shortest Distance with vertex 0 as the source:

Vertex 0: 0
Vertex 1: 0
Vertex 2: 0
Vertex 3: 0

Shortest Distance with vertex 1 as the source:

Vertex 0: inf
Vertex 1: 0
Vertex 2: 0
Vertex 3: 0

Shortest Distance with vertex 2 as the source:

Vertex 0: inf
Vertex 1: inf
Vertex 2: 0
Vertex 3: 0

Shortest Distance with vertex 3 as the source:

Vertex 0: inf
Vertex 1: inf
Vertex 2: inf
Vertex 3: 0

Time Complexity: The time complexity of the above algorithm is O(V^3 + V*E) as Dijkstra’s Algorithm takes O(n^2) for adjacency matrix. Note that the above algorithm can be made more efficient by using adjacency list instead of the adjacency matrix to represent the Graph.



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