Algorithms for Informed Cows

Ming-Yang Kao and Michael L. Littman Department of Computer Science Duke University, Durham, NC 27708-0129 fkao, [email protected] Abstract We extend the classic on-line search problem known as the cow-path problem to the case in which goal locations are selected according to one of a set of possible known probability distributions. We present a polynomial-time linear-programming algorithm for this problem. Introduction In on-line search, a great deal of work has been carried out concerning how to minimize the worst-case performance of a search strategy, for example as compared to an omniscient searcher. In many applications, this notion of optimal performance is simply too conservative|the agent might know, based on its prior experience, that some locations are more promising as targets than others. In this paper, we address the problem of nding good search strategies given a particular kind of prior knowledge. The search scenario we consider is the classic cow-path problem (Baeza-Yates, Culberson, & Rawlins 1993), in which a cow stands on a path and wants to nd a pasture in which to feed (see Figure 1). There are n locations to either side of the cow: p n through p 1 on its left, and p1 through pn on its right. Exactly one location contains a pasture and the cow can identify a pasture only by standing in it. The prior knowledge the cow has is expressed in terms of discrete probability distributions over the 2n possible locations of the pasture. We assume the cow knows that the pasture location is distributed according to one of m possible probability distributions 1; : : : ; m, but it doesn't know which. The cow would like to follow a search pattern that minimizes the worst possible expected value of the ratio of the number of steps it takes to nd a pasture to the number of steps it would need if it knew the probability distribution in advance (i.e., that the pasture location is distributed according to i for some 1 i m). Thus, the cow is required to minimize its worst-case competitive ratio (Sleator & Tarjan 1985; Yao 1980). In the next section, we look at the case in which the cow knows the probability distribution of the passtart ?