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• W2951038604 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 $p > frac{1}{2}$ (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 $(1 - delta)frac{log_2 n}{1 - H(p)} + o(log n) + O(log^2 (1/delta))$ queries, and identifies the target correctly with probability at leas $1-delta$. 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 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 $log_2 n$ 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 semi-adaptive version, in which one may query $r$ nodes each in $k$ rounds, we show membership in $Sigma_{2k-1}$ in the polynomial hierarchy, and hardness for $Sigma_{2k-5}$." @default.
• W2951038604 created "2019-06-27" @default.
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• W2951038604 date "2015-03-03" @default.
• W2951038604 modified "2023-09-27" @default.
• W2951038604 title "Deterministic and Probabilistic Binary Search in Graphs" @default.
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