Analysis of the expected number of bit comparisons required by Quickselect

TL;DR

This paper analyzes the expected number of bit comparisons needed by Quickselect, providing exact and asymptotic formulas, and demonstrating that the expectation grows linearly with the number of keys, with detailed results for specific key ranks.

Contribution

It introduces exact and asymptotic formulas for bit comparisons in Quickselect, extending analysis from key comparisons to bit-level performance metrics.

Findings

Expected bit comparisons are asymptotically linear in number of keys.

Exact formulas for smallest and largest key selection.

Finite summation formulas for arbitrary rank key selection.

Abstract

When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank.