a random permutation of [1,2,3,4,5,6,7,8]. Determine the probability that exactly 12 comparisons are required by Merge Sort to sort the input array A. Clearly and carefully justify your ans

Prove Proposition: Top-down mergesort uses between ½ N lg N and N lg N compares to sort any array of length N.

Hi I was wondering if I could get help with this question. I know that the first five  will make the largest sum of 9 but I don't where to go from there  Thanks

Can someone help me with this algorithm? No coding needed

we'll be working with the following set of nine numbers for binary searches set below:4, 10, 15, 18, 23, 26, 30, 34, 40Our objective is to pinpoint the number 26 within this array.The initial segment will comprehensively explain how to conduct a linear search to locate the number 26 within this array.

We apply the binary search on a 17-element ordered array. Assume that a givenkey appears in the array, and it is between the 5th element and the 13th element(inclusive). How many comparisons do we need for the average-case efficiency?

Let M(n) be the minimum number of comparisons needed to sort an array A with exactly n ele- ments. For example, M(1) = 0, M(2) = 1, and M(4) = 4. If n is an even number, clearly explain why M(n) = 2M(n/2) + n/2.

Let M[1 .. n] be an array, where each entry contains either a positive or a negative integer. Assume that M is already sorted in decreasing order. We want to decide whether there exists an entry M[r] such that M[r] is equal to r.

Problem 2. Suppose we have an array A[1 : n] which consists of numbers {1,...,n} written in some arbitrary order (this means that A is a permutation of the set {1,...,n}). Our goal in this problem is to design a very fast randomized algorithm that can find an index i in this array such that A[i] mod 8 € {1,2}, i.e., the reminder of dividing A[i] by 8 is either 1 or 2. For simplicity, in the following, we assume that n itself is a multiple of 8 and is at least 8 (so a correct answer always exist). For instance, if n = 8 and the array is A = [8,7, 2,5, 4, 6,3, 1], we want to output either of indices 3 or 8. (a) Suppose we sample an index i from {1,...,n} uniformly at random. What is the probability that i is a correct answer, i.e., A[i] mod 8 E {1,2}? (b) Suppose we sample m indices from {1,...,n} uniformly at random and with repetition. What is the probability that none of these indices is a correct answer? Now, consider the following simple algorithm for this problem: Find-Index-1(A[1:…