Find the shortest distance between any pair of two different good nodes
Given a weighted undirected connected graph with N nodes and M edges. Some of the nodes are marked as good. The task is to find the shortest distance between any pair of two different good nodes.
Note: Nodes marked as yellow in the below examples are considered to be good nodes.
Output : 7 Explanation : Pairs of Good Nodes and distance between them are: (1 to 3) -> distance: 7, (3 to 5) -> distance: 9, (1 to 5) -> distance: 16, out of which 7 is the minimum. Input :
Output : 4
Approach: Let us start by thinking of an algorithm to solve a simpler version of the given problem wherein all edges are of weight 1.
- Pick a random good node and perform a BFS from this point and stop at the first level say which contains another good node.
- We know that the minimum distance between any two good nodes can’t be more than s. So we again take a good node at random which is not already taken before and perform a BFS again. If we don’t find any special node in s distance, we terminate the search. If we do, then we update the value of s and repeat the procedure with some other special node taken at random.
We can apply a similar algorithm when weights are multiple.
Below is the implementation of the above approach:
Time complexity : O(V + E)
Here V is the number of vertices and E is the number of edges in the graph.
Space complexity : O(V+E)
The space complexity is mainly for the adjacency list used to represent the graph and the priority queue used to process the vertices.
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