The average running time of an algorithm as a midpoint between fuzzy sets

In 2003, Nieto and Torres introduced the notions of segment and midpoint between fuzzy sets with the aim of giving applications to medicine [J.J. Nieto, A. Torres, Midpoints for fuzzy sets and their application in medicine, Artif. Intell. Med. 27 (2003) 81–101]. Since then the interest in the study of such concepts have grown significantly because of their applicability to model real problems where the solution can be associated with a range of “middle ways” between two given positions. Recently, J. Casasnovas and F. Roselló have generalized the previous work of Nieto and Torres, giving an explicit description of segments and midpoints between fuzzy sets for, among others, the well-known weighted maximum distance [J. Casasnovas, F. Roselló, Averaging fuzzy biopolymers, Fuzzy Sets Syst. 152 (2005) 139–158].On the other hand, in 1999 Romaguera and Schellekens introduced the theory of dual complexity spaces as a part of the development of a topological foundation for the complexity analysis of programs and algorithms [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311–322]. Later on, this complexity structure was extended by García-Raffi, Romaguera and Sánchez-Pérez in order to obtain a suitable framework for the complexity analysis of exponential time algorithms [L.M García-Raffi, S. Romaguera, E.A. Sánchez-Pérez, The supremum asymmetric norm on sequence algebras: A general framework to measure complexity spaces, Electronic Notes in Theoret. Comput. Sci. 74 (2003) 12]. Inspired by the fact that in this theory the role of complexity measure is carried out by an asymmetric distance, in this paper we propose an asymmetric version of the weighted maximum distance to apply the midset theory to complexity analysis. So concise descriptions of segments and midsets between fuzzy sets for the mentioned asymmetric distance are provided. As an application of the obtained results, we prove formally that the average running time, for the Largetwo algorithm, is a midpoint between the running time of computing of the best case and the worst case by means of several connections between our new weighted maximum distance and the complexity measure introduced by García-Raffi, Romaguera and Sánchez-Pérez.