A recursive algorithm is applied to some data A = (a₁,..., am) where m≥ 2. The running time T is characterised using the following recurrence equations: T(2) = c when the size of A is 2 T(m) = T(m-1) + 2c otherwise Determine the running time complexity of this algorithm.

Consider the following recursive algorithm Algorithm Q(n)// Input: a positive integer nif n == 1 return 1else return Q(n-1)+2*n-1 1) Set up a recurrence relation for this function’s value and solve it to determine what this algorithm computes. 2) Setup a recurrence relation for the number of multiplication made by this algorithm and solve it to find the total number of multiplications that are executed.

1. Determine the running time of the following algorithm. Write summations to represent loops and solve using bounding. Be sure to show work on both the upper bound and lower bound, justify the split, and check that the bounds differ by only a constant factor. Use asymptotic notation to write the answer. Func1(n) 1 2 3 4 5 6 7 8 9 10 11 S← 0; for i ←n to n² do for j← 1 to i do for k9n to 10n² do for mi to k end end end return (s); end s+s + i- j + k −m;

4. Practice with the iteration method. We have already had a recurrence relation ofan algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c.(a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.Answer:(b)  Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction.Then by finding out the eventual number of t and substituting it into your generalform of T(n), you get the O(·) notation of T(n).]

Build a recursion tree for the following recurrence equation and then solve for T(n)

PLEASE explain how to solve this STEP BY STEP

Consider an n by n matrix, where each of the n2 entries is a positive integer. If the entries in this matrix are unsorted, then determining whether a target number t appears in the matrix can only be done by searching through each of the n2 entries. Thus, any search algorithm has a running time of O(n²). However, suppose you know that this n by n matrix satisfies the following properties: • Integers in each row increase from left to right. • Integers in each column increase from top to bottom. An example of such a matrix is presented below, for n=5. 4 7 11 15 2 5 8 12 19 3 6 9 16 22 10 13 14 17 24 1 18 21 23 | 26 | 30 Here is a bold claim: if the n by n matrix satisfies these two properties, then there exists an O(n) algorithm to determine whether a target number t appears in this matrix. Determine whether this statement is TRUE or FALSE. If the statement is TRUE, describe your algorithm and explain why your algorithm runs in O(n) time. If the statement is FALSE, clearly explain why no…

Time comp.

Let T(n) be defined by the first-order linear recurrence T(n) = 2T(n-1) +8 Suppose it is given that T(2) = c. Compute T(0) by iterating backwards and express your answer in terms of c. T(0) =

5. You are given a set of n positive numbers A = {a₁,..., an} and a positive integer t. Design a dynamic programming algorithm running in O(nt) time that decides whether there exists a subset A' CA such that Σ x = t. Note that each element of A can be xЄA' used at most once. Is the run-time of your algorithm polynomial with respect to the size of the input?