Let A be an array of n integers, where n is divisible by 4. Describe an algorithm with O(n) time complexity to decide whether there exists an element that occurs more than n/4 times in A. (a) Show that if such an element exists in A, it must be either the (n/4)-th, or the (2n/4)-th, or the (3n/4)-th smallest elements. (b) Use part (a) to derive your algorithm.

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Counting sort

The counting sort algorithm sorts an array’s contents by counting the repetition of each element that occurs in the array. A counting sort algorithm is a stable algorithm that works the best when the range of elements in the array is small, say for example from 0 to 9. The count which is the number of repetitions of an element in the array is mapped to an index of an auxiliary array, which is then saved in an auxiliary array. The counting sort algorithm has a stable time complexity of the order of O(size of array + size of auxiliary array). Let’s discuss the algorithm in detail as follows.