Average-case results on heapsort

Abstract

A new algorithm for rearranging a heap is presented and analysed in the average case. The average case upper bound for deleting the maximum element of a random heap is improved, and is shown to be less than [logn]+0.299+M(n) comparisons, *) whereM(n) is between 0 and 1. It is also shown that a heap can be constructed using 1.650n+O(logn) comparisons with this algorithm, the best result for any algorithm which does not use any extra space. The expected time to sortn elements is argued to be less thann logn+0.670n+O(logn), while simulation result points at an average case ofn log n+0.4n which will make it the fastest in-place sorting algorithm. The same technique is used to show that the average number of comparisons when deleting the maximum element of a heap using Williams' algorithm for rearrangement is 2([logn]−1.299+L(n)) whereL(n) also is between 0 and 1, and the average cost for Floyd-Williams Heapsort is at least 2nlogn−3.27n, counting only comparisons. An analysis of the number of interchanges when deleting the maximum element of a random heap, which is the same for both algorithms, is also presented.