The Cost Distribution of Queue-Mergesort, Optimal Mergesorts, and Power-of-2 Rules

Abstract

Queue-mergesort is introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queue-mergesort, including the best, average, and variance cases. The asymptotic normality of its cost is also established under the uniform permutation model. We address the corresponding optimality problems and we show that if we fix the merging scheme then the optimal mergesort as far as the average number of comparisons is concerned is to divide as evenly as possible at each recursive stage (top-down mergesort). On the other hand, the variance of queue-mergesort reaches asymptotically the minimum value. We also characterize a class of mergesorts with the latter property. A comparative discussion is given on the probabilistic behaviors of top-down mergesort, bottom-up mergesort, and queue-mergesort. We derive an “invariance principle” for asymptotic linearity of divide-and-conquer recurrences based on general “power-of-2” rules of which the underlying dividing rule of queue-mergesort is a special case. These analyses reveal an interesting algorithmic feature for general power-of-2 rules.