QuickHeapsort: Modifications and Improved Analysis

Abstract

QuickHeapsort is a combination of Quicksort and Heapsort. We show that the expected number of comparisons for QuickHeapsort is always better than for Quicksort if a usual median-of-constant strategy is used for choosing pivot elements. In order to obtain the result we present a new analysis for QuickHeapsort splitting it into the analysis of the partition-phases and the analysis of the heap-phases. This enables us to consider samples of non-constant size for the pivot selection and leads to better theoretical bounds for the algorithm. Furthermore, we introduce some modifications of QuickHeapsort. We show that for every input the expected number of comparisons is at most $n\log _{2}n - 0.03n + o(n)$ for the in-place variant. If we allow n extra bits, then we can lower the bound to $n\log _{2} n -0.997 n+ o (n)$. Thus, spending n extra bits we can save more that 0.96n comparisons if n is large enough. Both estimates improve the previously known results. Moreover, our non-in-place variant does essentially use the same number of comparisons as index based Heapsort variants and Relaxed-Weak-Heapsort which use $n\log _{2}n -0.9 n+ o (n)$ comparisons in the worst case. However, index based Heapsort variants and Relaxed-Weak-Heapsort require ${\Theta }(n\log n)$ extra bits whereas we need n bits only. Our theoretical results are upper bounds and valid for every input. Our computer experiments show that the gap between our bounds and the actual values on random inputs is small. Moreover, the computer experiments establish QuickHeapsort as competitive with Quicksort in terms of running time.