Minimum spanning tree cost of given Graphs
Given an undirected graph of V nodes (V > 2) named V1, V2, V3, …, Vn. Two nodes Vi and Vj are connected to each other if and only if 0 < | i – j | ≤ 2. Each edge between any vertex pair (Vi, Vj) is assigned a weight i + j. The task is to find the cost of the minimum spanning tree of such graph with V nodes.
Input: V = 4
Input: V = 5
Starting with a graph with minimum nodes (i.e. 3 nodes), the cost of the minimum spanning tree will be 7. Now for every node i starting from the fourth node which can be added to this graph, ith node can only be connected to (i – 1)th and (i – 2)th node and the minimum spanning tree will only include the node with the minimum weight so the newly added edge will have the weight i + (i – 2).
So addition of fourth node will increase the overall weight as 7 + (4 + 2) = 13
Similarly adding fifth node, weight = 13 + (5 + 3) = 21
For nth node, weight = weight + (n + (n – 2)).
This can be generalized as weight = V2 – V + 1 where V is the total nodes in the graph.
Below is the implementation of the above approach:
- Time Complexity: O(1)
- Auxiliary Space: O(1)