Optimizing two-sequence functionals in competitive analysis

Abstract

The efficiency of an on-line motion planning strategy often is measured by a constant competitive factor C. Competitivity means that the cost of a C-competitive on-line strategy with incomplete information is only C times worse than the optimal offline solution under full information. If a strategy is represented by an infinite sequence X=f 1,f 2,… of steps or values, the problem of finding a strategy with minimal C often results in minimizing functionals F k in X. For example $F_{k}(f_{1},f_{2},\ldots):=\frac{\sum_{i=1}^{k+1}f_{i}}{f_{k}}$ represents a functional for the 2-ray search problem. There are two main paradigms for finding an optimal sequence f 1,f 2,… that minimizes F k for all k. Namely, optimality of the exponential function and equality approach. If the strategy has to be defined by more than one interacting sequence both approaches may fail. In this paper we show that for such more sophisticated situations a combination of the paradigms is a good choice. As an example we consider an extension of the 2-ray search problem that can be formalized by two sequences.