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.
翻译:偏差 $ mathcal{P} 美元, 加权图形 $G$, 如果每个组群的直径最多为$\Delta$, 半径的每个球 $\Delta/\gma美元, 最多为$\tau$ 。 同样, $\mathcal{P} 美元是$(gmas,\ta,\Delta,\Delta), 如果对球来说, 如果每个组组群的直径最多为$(sgma,\Dau,\Delta) 美元, 则每条最短的直线长度为$\Delta/\gma$, 如果每个组群集的直径直径为$(sgmam,\ta/ta) 美元, 直径(Star_D), 直径的直径直路径(O_D) 和直径直径方的直径解(G_D美元), 直径直径方块的直径解,, 直径方块块=D, 直径解所有的直径解, 。