An Efficient Noisy Binary Search in Graphs via Median Approximation
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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^-2log 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^-1nlog n) per a single query. The second strategy is dedicated to graphs of small diameter D and maximum degree Δ and has the average complexity of O(n+ε^-2DΔlog 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^-2log n).
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