Suppose you are given a directed graph G = (V,E) with costs on the edges ce for e E and a sink t (costs may be negative). Assume that you also have finite values d(v) for v V. Someone claims that, for each node v V, the quantity d(v) is the cost of the minimum-cost path from node v to the sink t.

(a) Give a linear-time algorithm (time O(m) if the graph has m edges) that verifies whether this claim is correct.

(b) Assume that the distances are correct, and d(v) is finite for all v V. Now you need to compute distances to a different sink t. Give an O(m log n) algorithm for computing distances d(v) for all nodes v V to the sink node t. (Hint: It is useful to consider a new cost function defined as follows: for edge e = (v, w), let ce = ce d(v) + d(w). Is there a relation between costs of paths for the two different costs c and c?)