Cow-path Problem Research Articles

A nearly tight lower bound for the d-dimensional cow-path problem

In the d-dimensional cow-path problem, a cow living in Rd must locate a (d−1)-dimensional hyperplane H whose location is unknown. The only way that the cow can find H is to roam Rd until it intersects H. If the cow travels a total distance s to locate a hyperplane H whose distance from the origin was r≥1, then the cow is said to achieve competitive ratio s/r.It is a classic result that, in R2, the optimal (deterministic) competitive ratio is 9. In R3, the optimal competitive ratio is known to be at most ≈13.811. But in higher dimensions, the asymptotic relationship between d and the optimal competitive ratio remains an open question. The best upper and lower bounds, due to Antoniadis et al., are O(d3/2) and Ω(d), leaving a gap of roughly d. In this note, we achieve a stronger lower bound of Ω˜(d3/2).

Online searching with turn cost

We consider the problem of searching for an object on a line at an unknown distance OPT from the original position of the searcher, in the presence of a cost of d for each time the searcher changes direction. This is a generalization of the well-studied linear-search problem. We describe a strategy that is guaranteed to find the object at a cost of at most 9 · OPT + 2 d , which has the optimal competitive ratio 9 with respect to OPT plus the minimum corresponding additive term. Our argument for upper and lower bound uses an infinite linear program, which we solve by experimental solution of an infinite series of approximating finite linear programs, estimating the limits, and solving the resulting recurrences for an explicit proof of optimality. We feel that this technique is interesting in its own right and should help solve other searching problems. In particular, we consider the star search or cow-path problem with turn cost, where the hidden object is placed on one of m rays emanating from the original position of the searcher. For this problem we give a tight bound of ( 1 + 2 m m / ( m - 1 ) m - 1 ) OPT + m ( ( m / ( m - 1 ) ) m - 1 - 1 ) d . We also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line.

The CNN problem and other k-server variants

We study several interesting variants of the k-server problem. In the CNN problem, one server services requests in the Euclidean plane. The difference from the k-server problem is that the server does not have to move to a request, but it has only to move to a point that lies in the same horizontal or vertical line with the request. This, for example, models the problem faced by a crew of a Certain News Network trying to shoot scenes on the streets of Manhattan from a distance; for any event at an intersection, the crew has only to be on a matching street or avenue. The CNN problem contains as special cases two important problems: the BRIDGE problem, also known as the cow-path problem, and the weighted 2-server problem in which the 2 servers may have different speeds. We show that any deterministic online algorithm has competitive ratio at least 6+ 17 . We also show that some successful algorithms for the k-server problem fail to be competitive. In particular, no memoryless randomized algorithm can be competitive. We also consider another variant of the k-server problem, in which servers can move simultaneously, and we wish to minimize the time spent waiting for service. This is equivalent to the regular k-server problem under the L ∞ norm for movement costs. We give a 1 2 k(k+1) upper bound for the competitive ratio on trees.

On-line parallel heuristics, processor scheduling and robot searching under the competitive framework

In this paper we investigate parallel searches on m concurrent rays for a point target t located at some unknown distance along one of the rays. A group of p agents or robots moving at unit speed searches for t. The search succeeds when an agent reaches the point t. Given a strategy S the competitive ratio is the ratio of the time needed by the agents to find t using S and the time needed if the location of t had been known in advance. We provide a strategy with competitive ratio of 1+2( m/ p−1)( m/( m− p)) m/ p and prove that this is optimal. This problem has applications in multiple heuristic searches in AI as well as robot motion planning. The case p = 1 is known in the literature as the cow path problem.

Searching in an Unknown Environment: An Optimal Randomized Algorithm for the Cow-Path Problem

Searching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as thew-lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem; we give the first randomized algorithm in this paper. We show that our algorithm is optimal for two paths (w=2) and give evidence that it is optimal for larger values ofw. Subsequent to the preliminary version of this paper, Kaoet al.(in“Proceedings, 5th ACM–SIAM Symposium on Discrete Algorithm,” pp. 372–381, 1994) have shown that our algorithm is indeed optimal for allw⩾2. Our randomized algorithm gives expected performance that is almost twice as good as is possible with a deterministic algorithm. For the performance of our algorithm, we also derive the asymptotic growth with respect tow—despite similar complexity results for related problems, it appears that this growth has never been analyzed.