A partition $\mathcal{P}$ of a weighted graph $G$ is $(\sigma,\tau,\Delta)$-sparse if every cluster has diameter at most $\Delta$, and every ball of radius $\Delta/\sigma$ intersects at most $\tau$ clusters. Similarly, $\mathcal{P}$ is $(\sigma,\tau,\Delta)$-scattering if instead for balls we require that every shortest path of length at most $\Delta/\sigma$ intersects at most $\tau$ clusters. Given a graph $G$ that admits a $(\sigma,\tau,\Delta)$-sparse partition for all $\Delta>0$, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch $O(\tau\sigma^2\log_\tau n)$. Given a graph $G$ that admits a $(\sigma,\tau,\Delta)$-scattering partition for all $\Delta>0$, we construct a solution for the Steiner Point Removal problem with stretch $O(\tau^3\sigma^3)$. We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.

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分散和稀疏分区及其应用

加权图$G$的分区$\mathcal{P}$是$(\sigma,\tau,\Delta)$-sparse，如果每个簇的直径至多为$\Delta$，并且每个球的半径为$\ Delta/\sigma$ 最多与 $\tau$ 簇相交。类似地，$\mathcal{P}$ 是 $(\sigma,\tau,\Delta)$-散射，如果对于球，我们要求每条长度最多为 $\Delta/\sigma$ 的最短路径最多与 $\ 相交tau$ 簇。给定一个图 $G$，它承认所有 $\Delta>0$ 的 $(\sigma,\tau,\Delta)$-稀疏分区，Jia 等人。[STOC05] 为通用 Steiner 树问题（以及通用 TSP）构建了一个具有拉伸 $O(\tau\sigma^2\log_\tau n)$ 的解决方案。给定一个图 $G$ 承认所有 $\Delta>0$ 的 $(\sigma,\tau,\Delta)$-散射分区，我们构造了一个带有拉伸 $O(\tau ^3\sigma^3)$。