线性与周期域上连通性维护与点扩散的线性时间变换
Linear-Time Transformations Between Connectivity Maintenance and Points Spreading on Linear and Cyclic Domains
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中文总结 AI 辅助
本文针对直线与周期域，将连通性维护与点扩散问题在线性时间内相互归约，由此得到最小-求和连通性维护的求解时间，并扩展了归约的适用场景。
中文摘要 AI 辅助
给定直线或闭合周期上的n个点及阈值r>0，连通性维护问题是移动点使相邻点间的每个间隙不超过r，而点扩散问题要求每个间隙至少为r。Li和Wang[CCCG 2015; CGT 2025]给出了周期域上最小-最大点扩散的O(n)时间算法，Chen、Gu、Li和Wang[SWAT 2012; DCG 2013]给出了周期域上最小-最大连通性维护的O(n)时间算法，Ghadiri和Yazdanbod[CCCG 2016]给出了直线域上最小-求和点扩散的O(n log n)时间算法。本文证明，对于两种目标及直线、周期两种域，这两个问题可在线性时间内相互归约；由此，最小-求和连通性维护在两种域上均可在O(n log n)时间内求解；最后，将该归约扩展到初始顺序保持的直线上具有独立阈值的点。
英文摘要
Given $n$ points on a line or closed cycle and a threshold $r>0$, the connectivity-maintenance problem is to move the points so that every gap between consecutive points is at most $r$, whereas the points-spreading problem requires every gap to be at least $r$. Li and Wang [CCCG 2015; CGT 2025] and Chen, Gu, Li, and Wang [SWAT 2012; DCG 2013] gave $O(n)$-time algorithms for the cyclic versions of min-max points-spreading and min-max connectivity-maintenance, respectively. Ghadiri and Yazdanbod [CCCG 2016] gave an $O(n\log n)$-time algorithm for the linear version of min-sum points-spreading. In this paper, we show that the two problems can be reduced in linear time to each other for both objectives and on both linear and cyclic domains. As an implication, min-sum connectivity-maintenance is solvable in $O(n\log n)$ time on both domains. Finally, we extend the reduction to points on a line with individual thresholds when their initial order is preserved.
