Counting Zeros in Random Walks on the Integers and Analysis of Optimal Dual-Pivot Quicksort

TL;DR

This paper analyzes the average case performance of two dual-pivot quicksort variants, providing exact and asymptotic expressions for comparisons and exploring zeros of lattice paths, with a combinatorial identity proven.

Contribution

It introduces a precise average case analysis for dual-pivot quicksort variants and develops a novel approach using lattice path zeros and combinatorial identities.

Findings

Exact formulas for comparisons in dual-pivot quicksort

Asymptotic analysis of comparison counts

New combinatorial identity related to lattice paths

Abstract

We present an average case analysis of two variants of dual-pivot quicksort, one with a non-algorithmic comparison-optimal partitioning strategy, the other with a closely related algorithmic strategy. For both we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear, logarithmic, and constant terms. An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proven.