# Average Cost of QuickXsort with Pivot Sampling

## TL;DR

This paper provides a precise average-case analysis of QuickXsort with pivot sampling, extending existing methods and offering tight bounds that include linear terms, thereby improving understanding of its efficiency.

## Contribution

It offers a detailed recurrence solution for QuickXsort with pivot sampling, enabling accurate average cost calculations and extending analysis to QuickHeapsort and QuickMergesort.

## Key findings

- Derived exact average costs including linear terms.
- Extended analysis to QuickHeapsort and QuickMergesort.
- Provided tight upper bounds for these sorting methods.

## Abstract

QuickXsort is a strategy to combine Quicksort with another sorting method X, so that the result has essentially the same comparison cost as X in isolation, but sorts in place even when X requires a linear-size buffer. We solve the recurrence for QuickXsort precisely up to the linear term including the optimization to choose pivots from a sample of k elements. This allows to immediately obtain overall average costs using only the average costs of sorting method X (as if run in isolation). We thereby extend and greatly simplify the analysis of QuickHeapsort and QuickMergesort with practically efficient pivot selection, and give the first tight upper bounds including the linear term for such methods.