A General Framework for Searching on a Line

Abstract

Consider the following classical search problem: a target is located on the line at distance D from the origin. Starting at the origin, a searcher must find the target with minimum competitive cost. The classical competitive cost studied in the literature is the ratio between the distance travelled by the searcher and D. Note that when no lower bound on D is given, no competitive search strategy exists for this problem. Therefore, all competitive search strategies require some form of lower bound on D.

We develop a general framework that optimally solves several variants of this search problem. Our framework allows us to match optimal competitive search costs for previously studied variants such as: (1) where the target is fixed and the searcher’s cost at each step is a constant times the distance travelled, (2) where the target is fixed and the searcher’s cost at each step is the distance travelled plus a fixed constant (often referred to as the turn cost), (3) where the target is moving and the searcher’s cost at each step is the distance travelled.

Our main contribution is that the framework allows us to derive optimal competitive search strategies for variants of this problem that do not have a solution in the literature such as: (1) where the target is fixed and the searcher’s cost at each step is $\alpha _1 x+\beta _1$ for moving distance x away from the origin and $\alpha _2 x + \beta _2$ for moving back with constants $\alpha _1, \alpha _2, \beta _1, \beta _2$, (2) where the target is moving and the searcher’s cost at each step is a constant times the distance travelled plus a fixed constant turn cost. Notice that the latter variant can have several interpretations depending on what the turn cost represents. For example, if the turn cost represents the amount of time for the searcher to turn, then this has an impact on the position of the moving target. On the other hand, the turn cost can represent the amount of fuel needed to make an instantaneous turn, thereby not affecting the target’s position. Our framework addresses all of these variations.

This work was supported by FQRNT and NSERC.