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In 1994, S.G. Matthews introduced the notion of partial metric space in orderto obtain a suitable mathematical tool for program verification [Ann. New YorkAcad. Sci. 728 (1994), 183-197]. He gave an application of this new structureto parallel computing by means of a partial metric version of the celebratedBanach fixed point theorem [Theoret. Comput. Sci. 151 (1995), 195-205]. Lateron, M.P. Schellekens introduced the theory of complexity (quasi-metric) spacesas a part of the development of a topological foundation for the asymptoticcomplexity analysis of programs and algorithms [Elec- tronic Notes in Theoret.Comput. Sci. 1 (1995), 211-232]. The applicability of this theory to theasymptotic complexity analysis of Divide and Conquer algorithms was alsoillustrated by Schellekens. In particular, he gave a new proof, based on theuse of the aforenamed Banach fixed point theorem, of the well-known fact thatMergesort 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 toanalyze the asymptotic complexity of algorithms in the spirit of Schellekens.Specifically, we show that a slight modification of the well-known Bairepartial metric on the set of all words over an alphabet constitutes anappropriate tool to carry out the asymptotic complexity analysis of algorithmsvia fixed point methods without the need for assuming the convergence conditioninherent to the defini- tion of the complexity space in the Shellekensframework. Finally, in order to illustrate and to validate the developed theorywe apply our results to analyze the asymptotic complexity of Quicksort,Mergesort and Largesort algorithms.

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