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CSE 5311 Fall 1999

Quiz #2

(10 points.) Show how Quicksort can be made to run in O(n lg n) time in the worst case by making use of an order statistics algorithm.

Use the worst-case linear-time median finding algorithm to pick the pivot in the partitioning step of Quick sort. This guarantees an equal split in linear time, so the recurrence is T(n) = 2T(n/2) + Theta(n) = Theta(nlgn).

(10 points.) Consider the following variant of insertion sort: for 2$\leq$i$\leq$n, to insert the key A[i] among A[1] \(\leq \ldots \leq\)A[i-1], perform a binary search to find the correct position for A[i].

How many key comparisons would be performed in the worst case?

We perform $\leq$ lg i comparisons when inserting key i+1. The total number of comparisons in the worst case is \(\Sigma_{i=2}^n lg i \;=\; \Theta(nlgn)\).