Path Queries

We will run Dijkstra Algorithm for all vertices in this approach and store the minimum distance in an array. Then we can answer the queries using this distance array.
 

Dijkstra Algorithm is one of the most popular algorithms in graph theory. A single-source shortest path algorithm gives the shortest path length to all vertices from a given vertex known as the source vertex. It is a greedy algorithm that uses a priority queue(Binary Heap) to pick the shortest distance edges until the minimum distance of all vertices is calculated.

Algorithm:

• Dijkstra Algorithm:
  • We will define a function dijkstra(graph,n, source) where graph is the given graph,n is the number of vertices in the graph, and source is the source vertex.
  • Create an array dist with size n+1  to store the minimum distance of all vertices from source.
  •  Initialize all its values with maximum integer value because at the beginning. We assume that all vertices are unreachable.
  • Set dist[source] as 0.
  • We will now create a min-priority queue in which push elements in the form of [distance, vertex] and extract the minimum distance path in each step.
  • We will also create an array visited to track vertices whose minimum distance is already calculated.
  • Initialize visited array with 0 as no vertex is visited yet.
  • Insert [dist[source],source] into the priority queue.
  • While all vertices are not visited, do the following steps:
    • Delete the minimum distance vertex cur from the queue and mark it as visited.
    • Now traverse by all the vertices connected to cur and if the vertex index is not visited yet and check if dist[index] is greater than dist[cur]+graph[cur][index], then set dis[index] as dis[cur] + graph[cur][index] and push [dis[index],index] into the priority queue.
  • The minimum distance of all vertices from the source is calculated, return the dist array.
• Make an adjacency matrix graph, graph from the given edges.
• We will initialize an array dis of size (V+1)*(V+1) to store all pairs of the shortest distance.
• Now we will run this Dijkstra function V times. We will iterate for each vertex index from 1 to V.
  • Set dist as dijkstra(graph, V, index) and store the minimum distances of source index to all vertices from dist into dis[index].
  • The value dis[index1][index2] will represent the minimum distance between vertex index1 and index2.
• We will now answer queries by iterating through all queries.
  • For each query node1,node2 we will append dis[node1][node2] into ans array.
• We will return ans array.

O(V*(V+E)*log(V) + Q), where V and E are the numbers of vertices and edges in the graph, and Q is the number of queries.

The time complexity of insert, delete operation on priority queue is O(log(V)).

The time complexity of the Dijkstra Algorithm:

V times delete operation while extracting the minimum value vertex :O(V *log(V))

E times insert operation while traversing neighbor vertices: O(E*log(V))

So,the total time complexity of Dijkstra Algorithm is O((V+E)*log(V)).

We are running Dijkstra Algorithm V times and answering Q queries, and each query takes O(1) time. Hence the overall time complexity will be  O(V*(V+E)*log(V) +Q).

O(V^2), where V is the number of vertices in the graph.

The O(V^2) space is used to store the minimum distance between all pairs of vertices, and in total, there are V*V=V^2 pairs. The adjacency matrix also requires the O(V^2) space. Hence, the overall space complexity is O(V^2).