Question

Determine whether the following statements are true or false with justification.

(a) Let G be an undirected connected graph whose edge weights are in powers of 2(1,2,4,8,16... etc). Then G has a unique MST.

(b) Given an undirected unweighted graph G with n vertices and m edges and two distinct vertices s, t, the shortest s → t path can be computed in O(m + n) time.

Answer 1

(a)G be an undirected connected graph whose edge weights are in powers of 2(1,2,4,8,16... etc).Then G has a unique MST.
Graph G may contain two or more edges with same weight,which may lead to more than MST.
Hence above statement is false.

(b)Given an undirected unweighted graph G with n vertices and m edges and two distinct vertices s, t, the shortest s → t path can be computed in O(m + n) time.

Breadth First Search (BFS)can be used to find shortest path in undirected unweighted graph G for distinct vertices s and t.
Time complexity of BFS = O(V+E) where V is number of vertices and E is number of Edges.
Therefore Time Complexity = O(m+n).
Hence above statement is true.