Whereas the incremental algorithm uses induction, the divide$-$and$-$conquer alg orithm uses the technique of recursion, a powerful algorithm paradigm. The di vide$-$and$-$conquer paradigm partitions the problem into two parts, solves each of the parts recursively, and then "merges" the two solutions to obtain the full s olution.

$($a$.)$ Let $P_1$ and $P_2$ be two disjoint convex polygons with vertices in total. Give an time algorithm that computes the convex hull of $P_1{P}_2.$

$($b$.)$ Use the algorithm from part a to develop an time divide$-$and$-$conquer alg orithm to compute the convex hull of a set of points in the plane.