Abstract
In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by Emamjomeh-Zadeh et al. (in: Proceedings of the 48th annual ACM SIGACT symposium on theory of computing, STOC 2016, Cambridge, pp. 519–532, 2016) to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex—the “median”—which minimises a potential varPhi among the vertices of the candidates’ set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates’ set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential varGamma that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential varGamma that allows querying approximate medians.
Highlights
The classical binary search algorithm detects an unknown target t on a path with n vertices by asking at most log n queries to an oracle which always returns the direction from the queried vertex to t
The direction query considered in [13] either returns that the queried vertex q is the sought target t, or it returns an arbitrary direction from q to t, i.e. an arbitrary edge incident to q which lies on a shortest path from q to t
This paper resolves some of the open questions raised by Emamjomeh-Zadeh et al [13] and makes a first step towards understanding the query complexity of detecting two targets on graphs
Summary
The classical binary search algorithm detects an unknown target (or “treasure”) t on a path with n vertices by asking at most log n queries to an oracle which always returns the direction from the queried vertex to t. Recently the binary search algorithm with log n direction queries has been extended to arbitrary graphs by Emamjomeh-Zadeh et al [13] In this case there may exist multiple paths, or even multiple shortest paths form the queried vertex to t. 1 2 a direction to a shortest path from the queried vertex q to the target, and with probability 1 − p an arbitrary edge (possibly adversarially chosen) incident to q The study of this problem was initiated in [14], where Ω(log n) and O(log n) bounds on the number of queries were. It is worth noting that techniques similar to [13] were used to derive frameworks for robust interactive learning [11] and for adaptive hierarchical clustering [12]
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