The number of bit comparisons used by Quicksort: an average-case analysis

TL;DR

This paper analyzes the average number of bit comparisons used by Quicksort, introduces a new algorithm BitsQuick to reduce unnecessary bit comparisons, and compares bit-based and key-based analysis methods.

Contribution

It provides the first average-case analysis of bit comparisons in Quicksort and proposes BitsQuick to minimize redundant bit comparisons.

Findings

Bit comparisons introduce an extra logarithmic factor in Quicksort's complexity.

BitsQuick reduces bit comparisons to a constant order.

The analysis enables fair comparison between algorithms based on bit and key comparisons.

Abstract

The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Quicksort) are performed in terms of the number of key comparisons. We introduce the prospect of a fair comparison between algorithms of the two types by providing an average-case analysis of the number of bit comparisons required by Quicksort. Counting bit comparisons rather than key comparisons introduces an extra logarithmic factor to the asymptotic average total. We also provide a new algorithm, "BitsQuick", that reduces this factor to constant order by eliminating needless bit comparisons.