一种简单的图形重构算法
A Simple Algorithm for Graph Reconstruction
摘要：
我们如何有效地找到一个未知的图使用距离查询之间的顶点？我们假设未知图是连通的，无权的，有界度。目标是找到图中的每条边。该问题提出了一种基于多相Voronoi单元分解和$\tilde O(n^{3/2})$距离查询的重建算法。在我们的工作中，我们分析了一个简单的重建算法。我们表明，随机$Δ$ -正则图，我们的算法使用$\tilde O(n)$距离查询。作为副产品，我们可以重建这些图使用$O(\log^2 n)$查询到一个所有的距离oracle或$\tilde O(n)$查询到一个介数oracle，我们绑定这些图的度量维数的$\log^2 n$。我们的重建算法有一个非常简单的结构，是高度并行化。在一般的有界度图上，我们的重建算法具有次二次查询复杂度。
英文摘要：
How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using $\tilde O(n^{3/2})$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random $Δ$-regular graphs, our algorithm uses $\tilde O(n)$ distance queries. As by-products, we can reconstruct those graphs using $O(\log^2 n)$ queries to an all-distances oracle or $\tilde O(n)$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by $\log^2 n$. Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.
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