Randomized Binary and Tree Search Under Pressure

Abstract

We study a generalized binary search problem on the line and general trees. On the line (e.g., a sorted array), binary search finds a target node in O(log n) queries in the worst case, where n is the number of nodes. In time-constrained applications, we might only have time to perform a sub-logarithmic number of queries. In this case, it is impossible to guarantee that the target will be found regardless of its position. Our main result is the construction of a randomized strategy that maximizes the minimum (over the target’s position) probability of finding the target. Such a strategy provides a natural solution when there is no a priori (stochastic) information about the target’s position. As with regular binary search, we can find and run the strategy in O(log n) time (and using only O(log n) random bits). Our construction is obtained by reinterpreting the problem as a two-player zero-sum game and exploiting an underlying number theoretical structure.
For the more general case on trees, querying an edge returns the edge’s endpoint closest to the target. Given a bound k on the number of queries, we quantify a the-less-queries-the-better approach by defining a seeker’s profit p depending on the number of queries needed to locate the hider. For the linear programming formulation of the corresponding zero-sum game, we show that computing the best response for the hider (that is, the separation problem of the underlying dual LP) can be done in time O(n² 2^{2k}), where n is the size of the tree. This result allows us to compute a Nash equilibrium in polynomial time whenever k = O(log n). In contrast, computing the best response for the hider is NP-hard in general.