Aggregation of asymmetric distances in Computer Science

Abstract

In this paper we provide a general description of how to combine a collection (not necessarily finite) of asymmetric distances in order to obtain a single one as output. To this end we introduce the notion of asymmetric distance aggregation function that generalizes the well-known one for distance spaces given by Borsik and Doboš [J. Borsik, J. Doboš, On a product of metricspaces, Math. Slovaca 31 (1981) 193–205]. Among other results, a characterization of such functions is obtained in terms of monotony and subadditivity. Finally, we relate our results to Computer Science. In particular we show that the mathematical formalism based on complexity distances, which has been introduced by Romaguera and Schellekens [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topol. Appl. 98 (1999) 311–322] for the complexity analysis of programs and algorithms, can be obtained as a particular case of our new framework using appropriate asymmetric aggregation distance functions.

Introduction

Over the last years there has been a growing interest in the mathematical theory of information aggregation due to its wide range of applications to practical problems. In particular, for many processes that arise in applied sciences, such as image processing, decision making, control theory, medical diagnosis or biology, it is necessary to process incoming data that comes from sources of a different nature in order to obtain a conclusion. In such processes the pieces of information are symbolized via some numerical values. As a consequence the fusion methods that are based on numerical aggregation operators play a central role in the theory of information aggregation. A wide class of techniques of aggregation impose a constraint when selecting the most suitable aggregation operator for the problem to be solved. In general, this constraint consists of considering only those operators that provide the output data with the same properties as the inputs. An example of this type of situation is given when one wants to merge distances in order to obtain a new distance. Since the notion of distance plays a distinguished role in applied research, many authors have studied in depth which operators allow one to merge a collection of distances into a single one. In fact, in 1981 Borsik and Doboš profoundly studied the general problem of merging a collection of distances (not necessarily finite) into a single one [1]. Recently, Pradera et al. have provided, in the spirit of Borsik and Doboš, a general solution to the problem of merging data represented by means of a finite family of generalized distances and pseudodistances [17], [18], [19]. Several general techniques for merging a finite number of distances into another one have been studied by Casasnovas and Roselló in [3]. Specifically, they analysed the aggregation operators given by the weighted maximum and the weighted sum, and by the weighted Euclidean norm in order to apply some of their properties to the comparison of biological sequences. A related work, by the same authors, with applications to the diagnosis problem in medicine can be found in [4]. More recently, several connections between aggregation operators and distance functions have been given in [2], [15].

Asymmetric distances and other related structures have become efficient tools in some fields of Computer Science and in Bioinformatics. Metric tools based on asymmetric distances have been introduced and developed in order to provide an efficient framework to model processes, for instance, in complexity analysis of programs and algorithms [24], [21], [7], [8], [9], [10], [20], [16] and in logic programming [11], [22], [23]. In [25], [26], a natural correspondence has been obtained between similarity measures on biological (nucleotide or protein) sequences and asymmetric distances, giving practical applications to search in DNA and protein datasets. Motivated by the work of Rosselló and Casasnovas, a version of some results given in [4], [3] has been obtained for the case of aggregating asymmetric distances through the weighted sum in [5].

In this paper we focus our attention on providing a general description of how to combine a collection (not necessarily finite) of asymmetric distances in order to obtain a single one as output. We introduce the notion of asymmetric distance aggregation function which is a generalization of the one given by Borsik and Doboš in [1]. Moreover, we extend some results proved in the former reference in the context of asymmetric distances. In particular, we prove a characterization of asymmetric distance aggregation which differs to the given one for the classical case. More concretely it is shown that the class of such functions is exactly the set of all subadditive and monotone functions. Finally we connect the results obtained in our general framework with the complexity analysis of programs and algorithms in Computer Science. Hence we show that, in many instances, the asymmetric distances used in order to quantify the efficiency gained when an algorithm is substituted by another one can be retrieved by means of a family of distinguished asymmetric distance aggregation functions.

This document is organized as follows. Section 2 is devoted to the introduction of the basics of asymmetric distances, as well as a detailed exposition of the mathematical foundations, in the sense of Romaguera and Schellekens, of complexity analysis of programs and algorithms. In Section 3, we investigate the general problem of asymmetric distance aggregation, and in order to illustrate the obtained results we also show some connections between the developed theory and Computer Science in Section 4.