SIAM Journal on Computing, Volume 50, Issue 2, Page 662-673, January 2021.
Given $p$ node pairs in an $n$-node graph, a distance preserver is a sparse subgraph that agrees with the original graph on all of the given pairwise distances. We prove the following bounds on the number of edges needed for a distance preserver: 1. Any $p$ node pairs in a directed weighted graph have a distance preserver on $O(n + n^{2/3} p)$ edges. 2. Any $p = \Omega(\frac{n^2}{{\tt RS}(n)})$ node pairs in an undirected unweighted graph have a distance preserver on $O(p)$ edges, where ${\tt RS}(n)$ is the Ruzsa--Szemerédi function from combinatorial graph theory. 3. As a lower bound, there are examples where one needs $\omega(\sigma^2)$ edges to preserve all pairwise distances within a subset of $\sigma = o(n^{2/3})$ nodes in an undirected weighted graph. If we additionally require that the graph is unweighted, then the range of this lower bound falls slightly to $\sigma \le n^{2/3 - o(1)}$.

中文翻译：

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线性尺寸距离保持器的新结果

SIAM Journal on Computing，第 50 卷，第 2 期，第 662-673 页，2021 年 1 月。
给定 $n$-node 图中的 $p$ 节点对，距离保持器是一个稀疏子图，它在所有给定的成对距离上与原始图一致。我们证明了距离保持器所需边数的以下界限： 1. 有向加权图中的任何 $p$ 节点对在 $O(n + n^{2/3} p)$ 边上都有一个距离保持器. 2. 无向无权图中的任何 $p = \Omega(\frac{n^2}{{\tt RS}(n)})$ 节点对在 $O(p)$ 边上都有一个距离保持器，其中 $ {\tt RS}(n)$ 是来自组合图论的 Ruzsa--Szemerédi 函数。3. 作为下界，有些例子需要 $\omega(\sigma^2)$ 边来保留 $\sigma = o(n^{2/3})$ 节点子集中的所有成对距离无向加权图。如果我们另外要求图是未加权的，