How to play hot and cold

Abstract

Suppose we are searching for a target point t in a certain restricted search space. To pinpoint the location of t, we issue query points $q_1,\dots ,q_n$ in the search space. As a response, we obtain an ordering of the query points by distance to t. This restricts possible locations of t. In this paper, we consider the case where the search space is the unit interval $[0,1]$. 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 $\mathrm{\Theta }(n^2)$, which is at most a factor two from optimal if all query points have to be generated at once. With query points generated one by one depending on the response received on previous query points, we achieve accuracy $\mathrm{\Omega }({2.29}^n)$, and prove that no strategy can achieve $\mathrm{\Omega }({3.66}^n)$. These strategies can be extended to higher dimensional cases, where the search space is the unit cube or unit ball.