Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element that splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Then
(A)
T(n) <= 2T(n/5) + n
(B)
T(n) <= T(n/5) + T(4n/5) + n
(C)
T(n) <= 2T(4n/5) + n
(D)
T(n) <= 2T(n/2) + n
Answer: (B)
Explanation:
For the case where n/5 elements are in one subset, T(n/5) comparisons are needed for the first subset with n/5 elements, T(4n/5) is for the rest of 4n/5 elements, and n is for finding the pivot. If there are more than n/5 elements in one set then other sets will have less than 4n/5 elements and time complexity will be less than T(n/5) + T(4n/5) + n because the recursion tree will be more balanced.
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