Suppose that we are given an array A with n keys and k inversions. Here, an inversion is de ned as a pair of entries that are out of order in the array. What is the running time of Insertion sort when it is used to sort A in Big Oh notation? Why?
Develop a O(n log n) algorithm for computing the number of inversions in a given array.
A permutation (or ranking) is an array of N integers where each of the integers between 0 and N-1 appears exactly once. The Kendall tau distance between two rankings is the number of pairs that are in di erent order in the two rankings. For example, the Kendall tau distance between 0 3 1 6 2 5 4 and 1 0 3 6 4 2 5 9 is four since the pairs 0-1, 3-1, 2-4 and 5-4 are in di erent relative order in the two rankings but all the other pairs are in the same relative order. Develop a O(n log n) algorithm for computing the Kendall tau distance between two rankings. (Hint: use the solution to the previous problem.)
Suppose we are given a sequence S of n elements, each of which is an integer in the range [0; n2 1]. Describe a simple method for sorting S in O(n) time.
Prove that it is impossible to develop a comparison-based implementation of the MinPQ ADT such that both insert and delete the minimum take only O(log log n) time.
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