Dynamic Programming: Odds & Ends

Max-Cost Paths in DAGS Let G = (V,E) be a weighted directed graph. Dijkstra’s algorithm constructs a shortest path tree rooted at (any vertex) s in O(|E| log |V |) time (or O(|E|+ |V | log |V |) with more sophisticated data structures) as long as the algorithm does not contain any negative cycles. Surprisingly, the corresponding max-cost problem, i.e., finding a maximum cost simple path tree rooted at s, is not known to be solvable in polynomial time. As we will now see, though, this problem is solvable using a dynamic programming approach in O(E| + |V |) time if G is a Directed Acyclic Graph (DAG).