Let G = (V, E) be an n-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices u and v is a common ancestor w of u and v such that no descendant of w has the same property. In this paper, we consider the problem of computing an LCA, if any, for all
Consider a generalization of the classical binary search problem in linearly sorted data to the graph-theoretic setting. The goal is to design an adaptive query algorithm, called a strategy, that identifies an initially unknown target vertex in a graph by asking queries. Each query is conducted as follows: the strategy selects a vertex q and receives a reply v: if q is the target, then v = q, and if q is not the target, then v is a neighbor of q that lies on a shortest path to the target. Furthermore, there is a noise parameter 0 ≤ p < 1 2 , which means that each reply can be incorrect with probability p. The optimization criterion to be minimized is the overall number of queries asked by the strategy, called the query complexity. The query complexity is well understood to be O(ε −2 log n) for general graphs, where n is the order of the graph and ε = 1 2 − p. However, implementing such a strategy is computationally expensive, with each query requiring possibly O(n 2 ) operations.In this work we propose two efficient strategies that keep the optimal query complexity. The first strategy achieves the overall complexity of O(ε −1 n log n) per a single query. The second strategy is dedicated to graphs of small diameter D and maximum degree ∆ and has the average complexity of O(n + ε −2 D∆ log n) per query. We stress out that we develop an algorithmic tool of graph median approximation that is of independent interest: the median can be efficiently approximated by finding a vertex minimizing the sum of distances to a randomly sampled vertex subset of size O(ε −2 log n).
This work revisits the multiplicative weights update technique (MWU) which has a variety of applications, especially in learning and searching algorithms. In particular, the Bayesian update method is a well known version of MWU that is particularly applicable for the problem of searching in a given domain. An ideal scenario for that method is when the input distribution is known a priori and each single update maximizes the information gain. In this work we consider two search domains -linear orders (sorted arrays) and graphs, where the aim of the search is to locate an unknown target by performing as few queries as possible. Searching such domains is well understood when each query provides a correct answer and the input target distribution is uniform. Hence, we consider two generalizations: the noisy search both with arbitrary and adversarial (i.e., unknown) target distributions.We obtain several results providing full characterization of the query complexities in the three settings: adversarial Monte Carlo, adversarial Las Vegas and distributional Las Vegas. Our algorithms either improve, simplify or patch earlier ambiguities in the literature -see the works of Emamjomeh-Zadeh et al. [STOC 2016], Dereniowski et. al. [SOSA@SODA 2019 and Ben-Or and Hassidim [FOCS 2008]. In particular, all algorithms give strategies that provide the optimal number of queries up to lower-order terms. Our technical contribution lies in providing generic search techniques that are able to deal with the fact that, in general, queries guarantee only suboptimal information gain.
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