Deterministic and probabilistic binary search in graphs

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 > 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), log 2 n queries are always sufficient. For general p, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1−δ) log 2 n 1−H(p) +o(log n)+O(log 2 (1/δ)) queries, and identifies the target correctly with probability at least 1−δ. Here, H(p) = −(p log p+(1−p) log(1−p)) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1median 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 log 2 n 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 semiadaptive version, in which one may query r nodes each in k rounds, we show membership in Σ 2k−1 in the polynomial hierarchy, and hardness for Σ 2k−5 .

Binary Search in Graphs Revisited

“…This information-theoretic lower bound of [14] was matched by an improved upper bound in [2]. The same matching bound was extended to general graphs in [13].…”

“…Only 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. 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.…”

“…Only 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. 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. The main idea of this algorithm follows again the same principle as for paths and trees: it always queries a vertex that is the "median" of the current candidates' set and any response to the query is enough to shrink the size of the candidates' set by a factor of at least 2.…”

“…an arbitrary edge incident to q which lies on a shortest path from q to t. The main idea of this algorithm follows again the same principle as for paths and trees: it always queries a vertex that is the "median" of the current candidates' set and any response to the query is enough to shrink the size of the candidates' set by a factor of at least 2. Defining what the "median" is in the case of general graphs now becomes more tricky: Emamjomeh-Zadeh et al [13] define the median of a set S as the vertex q that minimizes a potential function Φ, namely the sum of the distances from q to all vertices of S.…”

“…In one of the variations of this model [2,13,14], each query independently returns with probability p > 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 established for a path with n vertices.…”