Python Program for Dijkstra’s shortest path algorithm | Greedy Algo-7

Given a graph and a source vertex in the graph, find shortest paths from source to all vertices in the given graph.

Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree. Like Prim’s MST, we generate a SPT (shortest path tree) with given source as root. We maintain two sets, one set contains vertices included in shortest path tree, other set includes vertices not yet included in shortest path tree. At every step of the algorithm, we find a vertex which is in the other set (set of not yet included) and has a minimum distance from the source.

Below are the detailed steps used in Dijkstra’s algorithm to find the shortest path from a single source vertex to all other vertices in the given graph.
Algorithm
1) Create a set sptSet (shortest path tree set) that keeps track of vertices included in shortest path tree, i.e., whose minimum distance from source is calculated and finalized. Initially, this set is empty.
2) Assign a distance value to all vertices in the input graph. Initialize all distance values as INFINITE. Assign distance value as 0 for the source vertex so that it is picked first.
3) While sptSet doesn’t include all vertices
….a) Pick a vertex u which is not there in sptSet and has minimum distance value.
….b) Include u to sptSet.
….c) Update distance value of all adjacent vertices of u. To update the distance values, iterate through all adjacent vertices. For every adjacent vertex v, if sum of distance value of u (from source) and weight of edge u-v, is less than the distance value of v, then update the distance value of v.

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# Python program for Dijkstra's single 
# source shortest path algorithm. The program is 
# for adjacency matrix representation of the graph
  
# Library for INT_MAX
import sys
  
class Graph():
  
    def __init__(self, vertices):
        self.V = vertices
        self.graph = [[0 for column in range(vertices)] 
                      for row in range(vertices)]
  
    def printSolution(self, dist):
        print "Vertex tDistance from Source"
        for node in range(self.V):
            print node, "t", dist[node]
  
    # A utility function to find the vertex with 
    # minimum distance value, from the set of vertices 
    # not yet included in shortest path tree
    def minDistance(self, dist, sptSet):
  
        # Initilaize minimum distance for next node
        min = sys.maxint
  
        # Search not nearest vertex not in the 
        # shortest path tree
        for v in range(self.V):
            if dist[v] < min and sptSet[v] == False:
                min = dist[v]
                min_index = v
  
        return min_index
  
    # Funtion that implements Dijkstra's single source 
    # shortest path algorithm for a graph represented 
    # using adjacency matrix representation
    def dijkstra(self, src):
  
        dist = [sys.maxint] * self.V
        dist[src] = 0
        sptSet = [False] * self.V
  
        for cout in range(self.V):
  
            # Pick the minimum distance vertex from 
            # the set of vertices not yet processed. 
            # u is always equal to src in first iteration
            u = self.minDistance(dist, sptSet)
  
            # Put the minimum distance vertex in the 
            # shotest path tree
            sptSet[u] = True
  
            # Update dist value of the adjacent vertices 
            # of the picked vertex only if the current 
            # distance is greater than new distance and
            # the vertex in not in the shotest path tree
            for v in range(self.V):
                if self.graph[u][v] > 0 and sptSet[v] == False and \
                dist[v] > dist[u] + self.graph[u][v]:
                        dist[v] = dist[u] + self.graph[u][v]
  
        self.printSolution(dist)
  
# Driver program
g = Graph(9)
g.graph = [[0, 4, 0, 0, 0, 0, 0, 8, 0],
           [4, 0, 8, 0, 0, 0, 0, 11, 0],
           [0, 8, 0, 7, 0, 4, 0, 0, 2],
           [0, 0, 7, 0, 9, 14, 0, 0, 0],
           [0, 0, 0, 9, 0, 10, 0, 0, 0],
           [0, 0, 4, 14, 10, 0, 2, 0, 0],
           [0, 0, 0, 0, 0, 2, 0, 1, 6],
           [8, 11, 0, 0, 0, 0, 1, 0, 7],
           [0, 0, 2, 0, 0, 0, 6, 7, 0]
          ];
  
g.dijkstra(0);
  
# This code is contributed by Divyanshu Mehta

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Output:

Vertex tDistance from Source
0 t 0
1 t 4
2 t 12
3 t 19
4 t 21
5 t 11
6 t 9
7 t 8
8 t 14

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