概率树嵌入的高效构造
Efficient Construction of Probabilistic Tree Embeddings
摘要：
在本文中，我们描述了一个算法，嵌入一个图度量$(V,d_G)$上的无向加权图$G=(V,E)$到树度量$(T,D_T)$的分布，使每对$u,v\in V$，$d_G(u,v)\leq d_T(u,v)$和${\bf{E}}_{T}[d_T(u,v)]\leq O(\log n)\cdot d_G(u,v)$。这样的嵌入在设计快速近似算法方面非常有用，因为图上的许多困难问题在树实例上很容易解决。对于具有$n$顶点和$m$边的图，我们的算法以高概率在$O(m\log n)$时间内运行，这改进了Mendel等人所给出的$O(m\log^3 n)$的上界。2009年该算法的核心是一种新的近似单源最短路径算法，该算法采用一种新的数据结构--桶树结构来实现优先级队列。该算法有三个属性：它只需要线性时间在输入图中的边的数量;计算的距离具有距离保持属性;当计算$k$ -从源最近的顶点的最短路径时，它只需要访问这些顶点和它们的边列表。这些属性对于保证正确性和规定的时间范围是必不可少的。使用这个最短路径算法，我们展示了如何生成一个中间结构，输入图的近似优势序列，在$O(m \log n)$时间，并进一步提出了一个简单而有效的算法转换这个序列到树嵌入在$O(n\log n)$时间，都具有很高的概率。结合这三个子程序给出了算法的规定时间范围。然后，我们表明，这种有效的建设，可以促进一些应用。我们证明了FRT树（生成的树嵌入）是具有渐近紧界的Ramsey划分，从而可以加速一系列距离预言机的构造。
英文摘要：
In this paper we describe an algorithm that embeds a graph metric $(V,d_G)$ on an undirected weighted graph $G=(V,E)$ into a distribution of tree metrics $(T,D_T)$ such that for every pair $u,v\in V$, $d_G(u,v)\leq d_T(u,v)$ and ${\bf{E}}_{T}[d_T(u,v)]\leq O(\log n)\cdot d_G(u,v)$. Such embeddings have proved highly useful in designing fast approximation algorithms, as many hard problems on graphs are easy to solve on tree instances. For a graph with $n$ vertices and $m$ edges, our algorithm runs in $O(m\log n)$ time with high probability, which improves the previous upper bound of $O(m\log^3 n)$ shown by Mendel et al.\,in 2009. The key component of our algorithm is a new approximate single-source shortest-path algorithm, which implements the priority queue with a new data structure, the "bucket-tree structure". The algorithm has three properties: it only requires linear time in the number of edges in the input graph; the computed distances have a distance preserving property; and when computing the shortest-paths to the $k$-nearest vertices from the source, it only requires to visit these vertices and their edge lists. These properties are essential to guarantee the correctness and the stated time bound. Using this shortest-path algorithm, we show how to generate an intermediate structure, the approximate dominance sequences of the input graph, in $O(m \log n)$ time, and further propose a simple yet efficient algorithm to converted this sequence to a tree embedding in $O(n\log n)$ time, both with high probability. Combining the three subroutines gives the stated time bound of the algorithm. Then we show that this efficient construction can facilitate some applications. We proved that FRT trees (the generated tree embedding) are Ramsey partitions with asymptotically tight bound, so the construction of a series of distance oracles can be accelerated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
