How to Play Hot and Cold on a Line

Abstract

Suppose we are searching for a target point t in the unit interval. To pinpoint the location of t, we issue query points $q_1, \ldots , q_n \in [0,1]$. As a response, we obtain an ordering of the query points by distance to t. This restricts possible locations of t to a subinterval. We define the accuracy of a query strategy as the reciprocal of the size of the subinterval to which we can pinpoint t in the worst case. We describe a strategy with accuracy $\Theta (n^2)$, which is at most a factor two from optimal if all query points are generated at once. With query points generated one by one depending on the response received on previous query points, we achieve accuracy $\Omega (2.29^n)$, and prove that no strategy can achieve $\Omega (3.66^n)$.