Dijkstra’s algorithm allows us to find the shortest path between any two vertices of a graph.
It differs from the minimum spanning tree because the shortest distance between two vertices might not include all the vertices of the graph.
How Dijkstra’s Algorithm works
Dijkstra’s Algorithm works on the basis that any subpath
B -> D of the shortest path
A -> D between vertices A and D is also the shortest path between vertices B and D.
Djikstra used this property in the opposite direction i.e we overestimate the distance of each vertex from the starting vertex. Then we visit each node and its neighbors to find the shortest subpath to those neighbors.
The algorithm uses a greedy approach in the sense that we find the next best solution hoping that the end result is the best solution for the whole problem.
Example of Dijkstra’s algorithm
It is easier to start with an example and then think about the algorithm.
Djikstra’s algorithm pseudocode
We need to maintain the path distance of every vertex. We can store that in an array of size v, where v is the number of vertices.
We also want to be able to get the shortest path, not only know the length of the shortest path. For this, we map each vertex to the vertex that last updated its path length.
Once the algorithm is over, we can backtrack from the destination vertex to the source vertex to find the path.
A minimum priority queue can be used to efficiently receive the vertex with least path distance.
function dijkstra(G, S) for each vertex V in G <- infinite <- NULL If V != S, add V to Priority Queue Q <- 0 while Q IS NOT EMPTY U <- Extract MIN from Q for each unvisited neighbour V of U tempDistance <- distance[U] + edge_weight(U, V) if tempDistance < distance[V] <- tempDistance <- U return distance, previous
Code for Dijkstra’s Algorithm
The implementation of Dijkstra’s Algorithm in Python is given below. The complexity of the code can be improved, but the abstractions are convenient to relate the code with the algorithm.
/* Dijkstra's Algorithm in Python */ import sys /* Providing the graph */ vertices = [[0, 0, 1, 1, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0], [1, 1, 0, 1, 1, 0, 0], [1, 0, 1, 0, 0, 0, 1], [0, 0, 1, 0, 0, 1, 0], [0, 1, 0, 0, 1, 0, 1], [0, 0, 0, 1, 0, 1, 0]] edges = [[0, 0, 1, 2, 0, 0, 0], [0, 0, 2, 0, 0, 3, 0], [1, 2, 0, 1, 3, 0, 0], [2, 0, 1, 0, 0, 0, 1], [0, 0, 3, 0, 0, 2, 0], [0, 3, 0, 0, 2, 0, 1], [0, 0, 0, 1, 0, 1, 0]] /* Find which vertex is to be visited next */ def to_be_visited(): global visited_and_distance v = -10 for index in range(num_of_vertices): if visited_and_distance[index] == 0 and (v < 0 or visited_and_distance[index] <= visited_and_distance[v]): v = index return v num_of_vertices = len(vertices) visited_and_distance = [[0, 0]] for i in range(num_of_vertices-1): visited_and_distance.append([0, sys.maxsize]) for vertex in range(num_of_vertices): /* Find next vertex to be visited */ to_visit = to_be_visited() for neighbor_index in range(num_of_vertices): /* Updating new distances */ if vertices[to_visit][neighbor_index] == 1 and visited_and_distance[neighbor_index] == 0: new_distance = visited_and_distance[to_visit] + edges[to_visit][neighbor_index] if visited_and_distance[neighbor_index] > new_distance: visited_and_distance[neighbor_index] = new_distance visited_and_distance[to_visit] = 1 i = 0 /* Printing the distance */ for distance in visited_and_distance: print("Distance of ", chr(ord('a') + i), " from source vertex: ", distance) i = i + 1
Dijkstra’s Algorithm Complexity
O(E Log V)
where, E is the number of edges and V is the number of vertices.
Dijkstra’s Algorithm Applications
- To find the shortest path
- In social networking applications
- In a telephone network
- To find the locations in the map
Python Example for Beginners
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