单位圆盘图的平衡线分离子
Balanced Line Separators of Unit Disk Graphs
摘要：
我们证明了单位圆盘相交图的图分离定理的一个几何版本：对于平面中的任何$n$单位圆盘集合，存在一条线$\ell$，使得$\ell$与至多$O(\sqrt{(m+n)\log{n}})$圆盘相交，并且由$\ell$确定的每个半平面至多包含集合中的$2n/3$单位圆盘，其中$m$是相交圆盘对的数目.我们还表明，一个轴平行线相交$O(\sqrt{m+n})$磁盘存在，但每个半平面可能包含多达$4n/5$磁盘。我们给出了一个几乎紧的下限（亚对数因子），我们的方法，也表明，没有线分离器的次线性大小的$n$存在，当我们看磁盘的任意半径，即使当$m=0$。证明是建设性的，并建议在线性时间内运行的简单算法。实验结果表明，在随机情况下，该方法优于Fox和Pach（其分隔符大小为$O(\sqrt{m})$）方法.
英文摘要：
We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $\ell$ such that $\ell$ intersects at most $O(\sqrt{(m+n)\log{n}})$ disks and each of the halfplanes determined by $\ell$ contains at most $2n/3$ unit disks from the set, where $m$ is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting $O(\sqrt{m+n})$ disks exists, but each halfplane may contain up to $4n/5$ disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in $n$ exists when we look at disks of arbitrary radii, even when $m=0$. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size $O(\sqrt{m})$).
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。
