Prove that The vertices reached in each invocation of the recursive procedure from the constructor are in a strong component in a DFS of a digraph G where marked vertices are treated in reverse postorder provided by a DFS of the digraph's reverse G R (Kosaraju's algorithm).

Give a dynamic programming algorithm using the following optimal subproblem structureLCIS+(i; j) : is the longest closed to increasing subsequence that ends and XiXj , i.e., its last elementis Xj and its second to last element is Xi. Give a recursive formula, prove its correctness and write abottom-up implementation.