在线几何覆盖与穿刺
Online Geometric Covering and Piercing
AI总结：
研究在线几何穿刺问题，针对尺寸相近的胖对象和同位相似超立方体提出确定性在线算法并证明竞争比上界，同时给出下界并推广到任意凸对象的在线单位覆盖。
摘要：
我们研究穿刺集问题的在线版本：几何对象逐个到达，在线算法必须通过做出不可撤销的决策，为已经到达的对象维护一个有效的穿刺集。容易观察到，对于 $\mathbb{R}$ 中的区间，任何解决该问题的确定性算法的竞争比至少为 $Ω(n)$。本文考虑尺寸相近对象的穿刺集问题。我们针对 $\mathbb{R}^d$ 中尺寸相近的胖对象提出一种确定性在线算法。对于边长位于范围 $[1,k]$ 内的 $\mathbb{R}^d$ 中同位相似超立方体，我们提出一种竞争比至多为 $3^d\lceil\log_2 k\rceil+2^d$ 的确定性算法。最后，我们给出了 $\mathbb{R}^2$ 中尺寸相近的 $α$-胖对象以及 $\mathbb{R}^d$ 中同位相似超立方体的竞争比确定性下界。需要注意的是，穿刺一个凸对象的平移副本等价于单位覆盖问题，而该问题在在线设定中已得到充分研究。令人惊讶的是，当相应对象不是球或超立方体时，单位覆盖问题此前没有任何已知的竞争比上界。我们的结果为相应对象是 $\mathbb{R}^d$ 中任意凸对象时的单位覆盖问题给出了竞争比上界。
英文摘要：
We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in $\mathbb{R}$ has a competitive ratio of at least $Ω(n)$. This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in $\mathbb{R}^d$. For homothetic hypercubes in $\mathbb{R}^d$ with side length in the range $[1,k]$, we propose a deterministic algorithm having a competitive ratio of at most~$3^d\lceil\log_2 k\rceil+2^d$. In the end, we show deterministic lower bounds of the competitive ratio for similarly sized $α$-fat objects in $\mathbb{R}^2$ and homothetic hypercubes in $\mathbb{R}^d$. Note that piercing translated copies of a convex object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when the corresponding object is anything other than a ball or a hypercube. Our result yields an upper bound of the competitive ratio for the unit covering problem when the corresponding object is any convex object in $\mathbb{R}^d$.
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。
