Note: Online Searching / on the Optimality of the Geometric Sequences for the m Ray Search Online Searching

Abstract

I the recent paper (Jaillet and Stafford 2001), we consider the following search problem: Given m concurrent branches (each a copy of R+), a searcher, initially placed at the origin, has to find an “exit,” which is at an unknown real distance d 1 > 0 from the origin on one of the m concurrent branches. The problem is to find a strategy that minimizes the worst-case ratio between the total distance traveled and the distance of the exit from the origin. The starting point of our work is the paper of Baeza-Yates et al. (1993), which discusses strategies for a problem very similar to our unbounded m-concurrent branch problem, in which the exit is at an integer distance from the origin. When m = 2, they give a proof that the strategy of alternatively moving on each branch, each time doubling the previous distance is optimal (among “monotone-increasing” strategies) with a competitive ratio of 9. When m> 2, they propose the following strategy: Move in the integrally increasing powers of m/ m− 1 in a cyclic manner, visiting branches in the same order over and over again. They argue that, among all “monotone-increasing” cyclic strategies, this one is optimal. One contribution of our paper is to provide a rigorous proof that the strategies introduced in Baeza-Yates et al. (1993) are optimal among all possible strategies. This is achieved via the introduction of a general mathematical programming/duality framework. This framework also allows us to obtain many generalizations such as optimal search strategies under additional deterministic and/or probabilistic information about the target location. It was brought to our attention recently that strategies very similar to those discussed in Baeza-Yates et al. (1993) had been introduced and independently analyzed in Gal (1980). This previous work was unknown to us and was not cited in Baeza-Yates et al. (1993), Jaillet and Stafford (2001), or in some subsequent publications. The purpose of this note is to give proper credit to this previous work. Looking at very similar search problems as the ones described above, Gal (1980) analyses “doubly infinite” search trajectories, for which no initial break point exists (a break point of a trajectory is any point on one of the m branches, excluding the origin, where the searcher changes direction). In other words, in the approach used by Gal, every admissible search trajectory has to start by making an infinite number of small “oscillations” near 0. Using very powerful results on the properties of exponential functions for this type of minimax search problem, the author is able to show the optimality of the doubly infinite sequence m/m− 1 i − <i< with cyclic pattern, where is any positive constant. The author then argues that to obtain a “practical” solution that does have a first break point, one can obtain an -optimal strategy by simply ignoring all the break points before a certain instant t0 where t0 is very small. Alternatively, the author notes that one would have to assume that the exit cannot be hidden closer than from the origin, and then modify the optimal trajectory and define it to be m/m− 1 i 0 i< . In both cases, the optimality of the resulting sequence is not formally established, but the heuristic arguments can be made rigorous, as shown in the note below (Gal 2002).