$4.(20$ points$)$ Let $\backslash ($ G$=($V$,$ E$)\backslash )$ 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 $\backslash ($ G $\backslash ).$ If $\backslash ($ G $\backslash )$ is acyclic, your algorithm should say so$.$ Your algorithm should run in $\backslash ($ O$\backslash$left$($n$^\{3\}\backslash$right$)\backslash )$ time, where $\backslash ($ n $\backslash )$ is the number of vertices in $\backslash ($ G $\backslash ).$ Explain the correctness of your algorithm. Derive its running time.

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