Use divide-and-conquer algorithm for finding the position of the maximum element in an array of n numbers. Set up a recurrence relation for the above algorithm and solve it.

Solving recurrences Solve the following recurrence relations and give a Θbound for eachof them. Do not use the Master Theorem. You must use the recursion tree method. For each recurrence,state the branching factor, the height of the tree, the size of the subproblems at depth k, and the number of subproblems at depth k. You can draw the recursion tree to help find the solution, but do not include the tree in your submission as it will not be graded.(a) T (n) = 4T (n/3)+n(b) T (n) = 3T (n/3)+n(c) T (n) = 4T (n/4)+√n(d) T (n) = 16T (n/4)+n2(e) T (n) = 3T (n/4)+1

Use the Transform-and-Conquer algorithm design technique with Instance Simplification variant to design an O(nlogn) algorithm for the problem below. Show the pseudocode. Given a set S of n integers and another integer x, determine whether or not there exist two elements in S whose sum is exactly x.

For the algorthim write a recurrence for its runtime, use the recurrence tree method to solve the recurrence, and and find the tightest asymptotic upper bound on the runtime of the algorthim. Algorthim Problem: Algorithm H divides an instance of size n into 4 subproblems of size n/3 each, recursively solves each one, and then takes O(n^2) time to combine the solutions and output the answer.

Write a recurrence for this algorithm and solve it to obtain a tight upper bound on the worst case runtime of this algorithm. You can use any method you like for solving this recurrence.

Use a recursion tree to determine a good asymptotic upper bound on the recurrence T(n)=4T(n/2+2)+n. Use the substitution method to verify your answer.