$(20$ points$)$ Let $G=(V,E)$ be a directed graph with positive edge weights.

$($a$)(10$ points$)$ The cost of a cycle is the sum of the weights of edges on that cycle. A cycle is called shortest if its cost is the minimum possible, Design an algorithm to return the cost of the shortest cycle in $G.$ If $G$ is acyclic, your algorithm should say so$.$ Your algorithm should run in $O(n^3)$ time, where $n$ is the number of vertices in $G.$ Explain the correctness of your algorithm. Derive its running time.

$($b$)(10$ points$)$ Suppose that the edge weights in $G$ are integers from the given range $0,W.$ Describe an implementation of Dijkstra's algorithm that runs in $O((n+m)logW)$ time.