Anatole Beck

On the linear search problem

A man in an automobile searches for another man who is located at some point of a certain road. He starts at a given point and knows in advance the probability that the second man is at any given point of the road. Since the man being sought might be in either direction from the starting point, the searcher will, in general, have to turn around many times before finding his target. How does he search so as to minimize the expected distance travelled? When can this minimum expectation actually be achieved? This paper answers the second of these questions.

Yet more on the linear search problem

The linear search problem has been discussed previously by one of the present authors. In this paper, the probability distribution of the point sought in the real line is not known to the searcher. Since there is noa priori choice of distribution which recommends itself above all others, we treat the situation as a game and obtain minimax type solutions. Different minimaxima apply depending on the factors which one wishes to minimize (resp. maximize). Certain criteria are developed which help the reader judge whether the results obtained can be considered “good advice” in the solution of real problems analogous to this one.

More on the linear search problem

The linear search problem concerns a search made in the real line for a point selected according to a given probability distribution. The search begins at zero and is made by continuous motion with constant speed along the line, first in one direction and then the other. The problem is to search in such a manner that the expected time required for finding the point according to the chosen plan of search is a minimum. This plan of search is usually conceived of as having a first step, a second,etc., and in that case, this author has previously shown a necessary and sufficient condition on the probability distribution for the existence of a search plan which minimizes the expected searching time. In this paper, we define a notion of search in which there is no first step, but the steps are instead numbered from negative to positive infinity. These new rules change the problem, and under them, there is always a minimizing search procedure. In those cases which satisfy the earlier criterion, the solutions obtained are essentially the same as those obtained previously.

Rendezvous Search on the Line with More Than Two Players

Suppose n blind, speed one, players are placed by a random permutation onto the integers 1 to n , and each is pointed randomly to the right or left. What is the least expected time required form m ≤ n of them to meet together at a single point? If they must all use the same strategy we call this time the symmetric rendezvous value R n , m s ; otherwise the asymmetric value R n , m a . We show that R 3,2 a = 47/48, and that R n , n s is asymptotic to n /2. These results respectively extend those for two players given by Alpern and Gal (Alpern, S., S. Gal. 1995. Rendezvous search on the line with distinguishable players. SIAM J. Control Optim. 33 1270–1276.) and Anderson and Essegaier (Anderson, E. J., S. Essegaier. 1995. Rendezvous search on the line with indistinguishable players. SIAM J. Control Optim. 33 1637–1642.).

The return of the linear search problem

The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes.

Son of the linear search problem

I wish to find something which is located on a certain road. I start at a point on the road, but I do not know in which direction the object sought is to be found. Somehow, I must incorporate in my way of searching the possibility that it is either to the right or to the left. Thus, I must search first to the right, and then to the left, and then to the right again until it is found. What is a good way of conducting this search, and what is a bad way?This general problem can be phrased in many ways mathematically, some of which are answered in the papers in the bibliography. In this paper, we consider three well-known assumptions concerning thea priori guesses for the probability distribution on where the object is located. These concern uniform distribution on an interval, triangular distribution around the original point, and normal distribution about that point. The uniform distribution has a simple answer. For the triangular distribution, we obtain qualitative results and calculate approximate values for the turning points.

Asymmetric Rendezvous on the Line is a Double Linear Search Problem

We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL). This problem, previously studied in a paper of Alpern and Gal (1995), asks how two blind, speed one players placed a distance d apart on the line, can find each other in minimum expected time. The distance d is drawn from a known cumulative probability distribution G, and the players are faced in random directions. We show that the ARSPL is strategically equivalent to a new problem we call the double linear search problem (DLSP), where an object is placed equiprobably on one of two lines, and equiprobably at positions ±d. A searcher is placed at the origin of each of these lines. The two searchers move with a combined speed of one, to minimize the expected time before one of them finds the object. Using results from a concurrent paper of the first author and J. V. Howard (1998), we solve the DLSP (and hence the ARSPL) for the case where G is convex on its support, and show that the solution is that conjectured in a paper of Baston and Gal (1998).

The revenge of the linear search problem

The linear search problem is the name for several problems motivated by the same external reality. At times, the search for a goal can proceed in two (or more) directions. Looking in one direction is at the expense of time and effort, which can be used elsewhere. More specifically, it might actually be moving further from the goal. This is modeled by a physical search along an infinite straight line, where the object of the search might be in either direction. Faced with a (known or unknown) probability distribution, this paper attempts to minimize the expected loss, where the loss is a function of the time of the search and the location of the object. In this variant of the problem, known distributions are dealt with, and the loss function is a known power of the time spent.

The linear search problem rides again

The Linear Search Problem concerns a search for a point in the real line by continuous motion starting at 0. The optimal turning points for such a search under the hypothesis that the location of the target is distributed normally about 0 have been approximated by mechanical calculation, but no proof has been given that there is only a single minimizing strategy or that the numbers calculated do indeed approximate that strategy. Plausible arguments have been made before, both by these authors and others. In this paper, the plausible arguments are supplanted by mathematical proofs.

Pure Strategy Asymmetric Rendezvous on the Line with an Unknown Initial Distance

Suppose two blind agents with unit speed are placed a distanceH apart on an infinite line, and faced in random directions. Their initial distanceH is picked from a distributionF with finite mean µ. We present a pair of rendezvous strategies which do not depend on the distributionF and ensure a meeting in expected time less than 5:514µ. This improves the bound of 5:74µ given by Baston and Gal. Furthermore, the bound we give is best possible for strategies of our type.