$($Graph Algorithms$)$

$($a$)(10)$ Define the Minimum$-$Spanning tree problem and provide an algorithm which solves this problem.

$($b$)(10)$ For a given weighted graph $G=(V,E,w),$ where $n=|V|$ and $m=|E|,$ the naive implementation of Prim's algorithm runs in $O(nm)$ time via exhaustive search over the adjacency$-$list representation of the input $G.$ Describe how a data structure can be used to speed up its performance to $O(n+m)$ time.