Deterministic and Probabilistic Binary Search in Graphs

Ehsan Emamjomeh-Zadeh, David Kempe, Vikrant Singhal

Abstract:

We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node q, the algorithm learns either that q is the target, or is given an edge out of q that lies on a shortest path from q to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability p>12 (a known constant), and an (adversarial) incorrect one with probability 1−p.
Our main positive result is that when p=1 (i.e., all answers are correct), log2n queries are always sufficient. For general p, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1−δ)log2n1−H(p)+o(logn)+O(log2(1/δ)) queries, and identifies the target correctly with probability at leas 1−δ. Here, H(p)=−(plogp+(1−p)log(1−p)) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm.
Even for p=1, we show several hardness results for the problem of determining whether a target can be found using K queries. Our upper bound of log2n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query r nodes each in k rounds, we show membership in Σ2k−1 in the polynomial hierarchy, and hardness for Σ2k−5.