Question

9-2. For each of the following problems: design a reduction algorithm a hash table or sorting algorithm that solves the problem; describe your algorithm with clear pseudocode; and prove the time efficiency class of your algorithm. duplicate search problem input: a vector V of comparable objects output: an element of V that appears more than once in V, or None if no such element exists

Answer 1

We will reduce this problem to sorting problem where we sort all the entries of vector V of comparable objects into sorted array and then determine whether there are two consecutive indices having same value. If there are two consecutive indices having same value, then there exist an element of V appearing more than once in V, otherwise not.

ALGORITHM(V, n)

1. Given the vector V of size n, store the element in V in array A in sorted order by using HEAPSORT algorithm.

2. For i = 1 to n-1:-

3..........if A[i] == A[i+1]:- //This means element at index i is repeated, hence return True

4................Return True

5. Return False //If now such index is found till the end, then return false, since no element is repeated.

Hence this procedure require to perform sorting which takes O(n log n) time if MERGESORT is used and then iterating through all the indices will take O(n) time. Hence total time complexity will be O(n log n) + O(n) = O(n log n).

Please comment for any clarification.