Computer Algorithm question

8) Give an algorithm for building a heap in O(n) 9) Prove the algorithm given in 8) runs in O(n) time.

10) What is the asymptotic runtime of an algorithm represented by the following recurrence equation?

11) Suppose you have the following priority queue implemented as a (max) heap. What will the heap look like when the max node is removed and the heap is readjusted? Assume on each heapify operation the largest child node is selected for the exchange with the parent node 20 18 15 12 17 14 12) Write an algorithm and give the best and worst case run time for randomized Quicksort ime for the given algorithm 14) Give an example of a divide and conquer algorithm. Give the worst case asymptotic run time for the given algorithm