论文标题
接近线性$ \ VAREPSILON $ - 平面图的emulators
Near-Linear $\varepsilon$-Emulators for Planar Graphs
论文作者
论文摘要
我们研究距离的顶点稀疏，在平面图的设置中：给定平面图$ g $（具有边缘的重量）和$ k $终端顶点的子集，目标是构建一个$ \ varepsilon $ emulator，这是一个小平面图$ g'$，其中包含端子和范围$ 1+1+1+1+1+1+1+1+1+1+1+1+serves upers $ 1+1+1+serves upers。我们为接近线性尺寸$ \ tilde o（k/\ varepsilon^{o（1）}）$的平面图构建了第一个$ \ varepsilon $ - emulators。就$ k $而言，这是对张，goranci和henzinger先前的二次上限的巨大改进，并在已知的二次二次下限范围以下（当$ \ varepsilon = 0 $时）中断了。此外，我们的模拟器可以在（接近）线性时间中计算，从而导致$（1+ \ varepsilon）$ - 近似算法，用于平面图上的基本优化问题，包括多个源最短路径，包括多个源的$（S，T）$ - $（S，T）$ - 切割，图形直径和动态距离。
We study vertex sparsification for distances, in the setting of planar graphs with distortion: Given a planar graph $G$ (with edge weights) and a subset of $k$ terminal vertices, the goal is to construct an $\varepsilon$-emulator, which is a small planar graph $G'$ that contains the terminals and preserves the distances between the terminals up to factor $1+\varepsilon$. We construct the first $\varepsilon$-emulators for planar graphs of near-linear size $\tilde O(k/\varepsilon^{O(1)})$. In terms of $k$, this is a dramatic improvement over the previous quadratic upper bound of Cheung, Goranci and Henzinger, and breaks below known quadratic lower bounds for exact emulators (the case when $\varepsilon=0$). Moreover, our emulators can be computed in (near-)linear time, which lead to fast $(1+\varepsilon)$-approximation algorithms for basic optimization problems on planar graphs, including multiple-source shortest paths, minimum $(s,t)$-cut, graph diameter, and dynamic distace oracle.
