Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths

TL;DR

This paper analyzes a variant of dual-pivot quicksort, proving its optimality in minimizing key comparisons and providing exact and asymptotic expressions for its expected comparisons, using lattice path analysis and combinatorial identities.

Contribution

It establishes the optimality of a dual-pivot quicksort strategy and develops exact and asymptotic analysis methods involving lattice path zeros.

Findings

The partitioning strategy is proven to be optimal.

Exact formulas for expected comparisons are derived.

Asymptotic analysis includes linear, logarithmic, and constant terms.

Abstract

We present an average case analysis of a variant of dual-pivot quicksort. We show that the used algorithmic partitioning strategy is optimal, i.e., it minimizes the expected number of key comparisons. For the analysis, 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.