Optimal Sampling Strategies in Quicksort and Quickselect

Abstract

It is well known that the performance of quicksort can be improved by selecting the median of a sample of elements as the pivot of each partitioning stage. For large samples the partitions are better, but the amount of additional comparisons and exchanges to find the median of the sample also increases. We show in this paper that the optimal sample size to minimize the average total cost of quicksort, as a function of the size n of the current subarray size, is a • √ n + o(√ n). We give a closed expression for a, which depends on the selection algorithm and the costs of elementary comparisons and exchanges. Moreover, we show that selecting the medians of the samples as pivots is not the best strategy when exchanges are much more expensive than comparisons. We also apply the same ideas and techniques to the analysis of quickselect and get similar results.

The height of increasing trees

“…Thus, without any further work, we can conclude from Theorem 2 that H n ∼ log 2 n in probability. This yields a simple randomized way of asymptotically balancing a binary search tree, strengthening a result of Martínez and Roura (2001) who proved that the average depth is asymptotically log 2 n.…”

LazySorted: A Lazily, Partially SortedPythonList

“…This leads to pivots that are closer to the middle of the array, decreasing the total number of partitions that need be completed. Still greater improvement could perhaps be attained by using the optimal pivot selection strategies described by Martínez and Roura (2001).…”

On a multivariate contraction method for random recursive structures with applications to Quicksort

“…with X (1) , X (2) ∼ X being independent and independent of (A * 1 , A * 2 , b * ), where (A * 1 , A * 2 , b * ) is given by (33), (34) with V there being beta(t + 1, t + 1) distributed.…”

“…The asymptotic mean of C n is decreasing for t → ∞ whereas the mean of B n is increasing as t → ∞. For the identification of the optimal t ∈ N 0 such that the leading constant τ (t, w) in E W n = (1 + o(1))τ (t, w)n ln(n) is minimized for given w > 0 see Martínez and Roura [33], who also discuss the more delicate problem of allowing t to be dependent on n.…”