divide and conquer strategy to understand the binary search for a sorted list of n elements. Let T(n) denote the time complexity function for the binary search. Derive a recurrence relation for T(n) and solve for T(n) to get a close form expression.

Use a recursion tree to determine a good asymptotic upper bound on therecurrence T(n) = 3T(n/2) + n. Use the substitution method to prove your answer.

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.

Define the subproblem as follows: Let M(i) be the maximum sum of all contiguous subarrays ending at position i. Clearly M(1) = A[1]. What is the recurrence relation for computing M(i)? Use this to give a more efficient algorithm to compute OPT.

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.

Question 5 Refer to the following graph and choose the option that reflects the row for node 3 of the D(2) matrix generated by the Floyd-Warshall Algorithm. If you are solving this programmatically, be careful about zero-based indices in most languages. The graph below has 1-based indexing. 1 O 32042-8 3 O-42 327620 -5 Infinity O 87 Infinity 9 0 Infinity O 0204年1月8日 10 5 -8

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.