Abstract:
We consider the following natural generalization of Binary Search to arbitrary connected graphs: 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 > 1/2 (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), log2 n queries are always sufficient. For general p, we give an (information-theoretically optimal) algorithm that uses at most (1 - δ) log n/(1 - H(p)) + o(log n) queries, and identifies the target correctly with probability at least 1-δ. (Here, H(p) denotes the entropy.)
We show several hardness results for the problem of determining, even for p = 1, whether a target can always be found using K queries. Our upper bound of log2 n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis. We show that for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete.