Suppose we have two arrays A[1:n] and B[1:m] which are both in sorted order and are consisting of distinct numbers. Design an algorithm that given an integer 1≤r≤m+n, find the element with rank in the union of arrays A and B. Your algorithm should run in only O(log (n+m)) time. Example. Suppose A= [1,5,7,9] and B= [2,4,6,12] and so n=m= 4. Then, the answer to r= 3 is 4 and the answer to r= 7 is 9 because the union of arrays A and B in the sorted order is [1,2,4,5,6,7,9,12]. The answers should not be code/program. Rather, it should be some pseudocode and/or English explanation with proof of correctness that leads to the run time it asks for.

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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.