The Baire partial quasi-metric space: A mathematical tool for asymptotic complexity analysis in Computer Science

TL;DR

This paper introduces a modified Baire partial quasi-metric space as a new mathematical tool for analyzing the asymptotic complexity of algorithms, enabling fixed point methods without convergence assumptions.

Contribution

It proposes a new partial quasi-metric framework based on a modification of the Baire metric for asymptotic complexity analysis in algorithms.

Findings

Validated the approach on Quicksort, Mergesort, and Largesort algorithms.

Provided a fixed point analysis method for asymptotic complexity without convergence assumptions.

Demonstrated the applicability of the modified metric to classical sorting algorithms.

Abstract

In 1994, S.G. Matthews introduced the notion of partial metric space in order to obtain a suitable mathematical tool for program verification [Ann. New York Acad. Sci. 728 (1994), 183-197]. He gave an application of this new structure to parallel computing by means of a partial metric version of the celebrated Banach fixed point theorem [Theoret. Comput. Sci. 151 (1995), 195-205]. Later on, M.P. Schellekens introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms [Elec- tronic Notes in Theoret. Comput. Sci. 1 (1995), 211-232]. The applicability of this theory to the asymptotic complexity analysis of Divide and Conquer algorithms was also illustrated by Schellekens. In particular, he gave a new proof, based on the use of the aforenamed Banach fixed point theorem, of the well-known fact that Mergesort al- gorithm has optimal asymptotic average running time of computing. In this paper, motivated by the utility of partial metrics in Computer Science, we discuss whether the Matthews fixed point theorem is a suitable tool to analyze the asymptotic complexity of algorithms in the spirit of Schellekens. Specifically, we show that a slight modification of the well-known Baire partial metric on the set of all words over an alphabet constitutes an appropriate tool to carry out the asymptotic complexity analysis of algorithms via fixed point methods without the need for assuming the convergence condition inherent to the defini- tion of the complexity space in the Shellekens framework. Finally, in order to illustrate and to validate the developed theory we apply our results to analyze the asymptotic complexity of Quicksort, Mergesort and Largesort algorithms.