More on the linear search problem

Abstract

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.