Question

8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can be done in O(n) time where n is the number of vertices in V.

Answer 1

Given A is adjacent matrix of directed graph of V vertices.
(A)Algorithm to check if graph has a sink or not

We need to eliminate n – 1 non-sink vertices in O(V) time and check the remaining vertex for the sink property.

step-1 To eliminate vertices, we check whether a particular index (A[i][j]) in the adjacency matrix is a 1 or a 0.
(i)If it is a 0, it means that the vertex corresponding to index j cannot be a sink.
(ii)If the index is a 1, it means the vertex corresponding to i cannot be a sink. We keep increasing i and j in this fashion until either i or j exceeds the number of vertices.

step-2 keep doing step-1 until one vertex left.
At last we are left with only vertex i.
We now check for whether row i has only 0s and whether row j as only 1s except for A[i][i], which will be 0.