一种有效的基于中值近似的图中噪声二叉搜索算法
An Efficient Noisy Binary Search in Graphs via Median Approximation
摘要：
考虑将线性排序数据中的经典二分查找问题推广到图论设置。我们的目标是设计一个自适应的查询算法，称为一个战略，确定一个最初未知的目标顶点在一个图中的询问查询。每个查询如下进行：策略选择顶点$q$并接收应答$v$：如果$q$是目标，则为$v=q$，如果$q$不是目标，则$v$是位于到目标的最短路径上的$q$的邻居。此外，存在噪声参数$0\leq p<\frac{1}{2}$，这意味着每个应答可能以概率$p$不正确。要最小化的优化标准是策略所要求的查询总数，称为查询复杂度。对于一般图，查询复杂度被很好地理解为$O(\varepsilon^{-2}\log n)$，其中$n$是图的阶，而$\varepsilon=\frac{1}{2}-p$是图的阶。然而，实现这样的策略在计算上是昂贵的，每个查询可能需要$O(n^2)$操作。在这项工作中，我们提出了两个有效的策略，保持最佳的查询复杂度。第一种策略实现了每个查询的$O(\varepsilon^{-1}n\log n)$的总体复杂性。第二个策略是专门用于小直径$D$和最大程度$Δ$的图形，并具有$O(n+\varepsilon^{-2}DΔ\log n)$每个查询的平均复杂度。我们强调，我们开发了一个算法工具的图的中位数近似，是独立的兴趣：中位数可以有效地近似通过找到一个顶点最小化的距离的总和随机采样的顶点子集的大小$O(\varepsilon^{-2}\log n)$。
英文摘要：
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 $0\leq p<\frac{1}{2}$, 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 $O(\varepsilon^{-2}\log n)$ for general graphs, where $n$ is the order of the graph and $\varepsilon=\frac{1}{2}-p$. 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 $O(\varepsilon^{-1}n\log n)$ 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 $O(n+\varepsilon^{-2}DΔ\log n)$ 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 $O(\varepsilon^{-2}\log n)$.
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