arXiv 1912.033502019-12cs.DScs.CGcs.DMcs.GT

在线矢量平衡与几何离散

Online Vector Balancing and Geometric Discrepancy

Nikhil Bansal, Haotian Jiang, Sahil Singla, Makrand Sinha

备注：: Appears in STOC 2020

摘要：

我们考虑一个在线的矢量平衡问题，$T$矢量，选择从任意分布在$[-1,1]^n$，到达一个接一个，必须立即给出一个$\pm$标志。我们的目标是保持差异尽可能小。一个具体的例子是在线区间差异问题，其中T点在[0，1]中均匀采样，目标是立即对它们进行$\pm$着色，使得每个子区间保持接近平衡。由于随机着色会导致$Ω(T^{1/2})$差异，而离线边界是向量平衡的$Θ(\sqrt{n \log (T/n)})$和区间平衡的$1$，一个自然的问题是是否可以（几乎）匹配这些问题的在线设置中的离线边界。必须利用随机性，因为在最坏的情况下，已知任何在线算法的差异都是$Ω(T^{1/2})$。Bansal和Spencer最近证明了当每个坐标独立时的$O(\sqrt{n}\log T)$界。当坐标之间存在依赖关系时，问题变得更具挑战性，Jiang，Kulkarni和Singla最近的工作证明了这一点，该工作给出了在线间隔差异的非平凡$O(T^{1/\log\log T})$。虽然这比随机着色好，但离离线边界还很远。在这项工作中，我们引入了一个新的框架，用于在线向量平衡时，输入分布具有跨坐标的依赖关系。这使我们可以获得任意输入分布下的在线向量平衡的$poly(n, \log T)$界，以及在线间隔差异的$poly(\log T)$界。我们的框架足够强大，可以捕获其他经过充分研究的几何差异问题;例如，一个$poly(\log^d (T))$绑定的在线$d$维Tusnády的问题。一个关键的新的技术成分是一个{反浓度}不等式的两两不相关的随机变量的总和。

英文摘要：

We consider an online vector balancing question where $T$ vectors, chosen from an arbitrary distribution over $[-1,1]^n$, arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the discrepancy small as possible. A concrete example is the online interval discrepancy problem where T points are sampled uniformly in [0,1], and the goal is to immediately color them $\pm$ such that every sub-interval remains nearly balanced. As random coloring incurs $Ω(T^{1/2})$ discrepancy, while the offline bounds are $Θ(\sqrt{n \log (T/n)})$ for vector balancing and $1$ for interval balancing, a natural question is whether one can (nearly) match the offline bounds in the online setting for these problems. One must utilize the stochasticity as in the worst-case scenario it is known that discrepancy is $Ω(T^{1/2})$ for any online algorithm. Bansal and Spencer recently show an $O(\sqrt{n}\log T)$ bound when each coordinate is independent. When there are dependencies among the coordinates, the problem becomes much more challenging, as evidenced by a recent work of Jiang, Kulkarni, and Singla that gives a non-trivial $O(T^{1/\log\log T})$ bound for online interval discrepancy. Although this beats random coloring, it is still far from the offline bound. In this work, we introduce a new framework for online vector balancing when the input distribution has dependencies across coordinates. This lets us obtain a $poly(n, \log T)$ bound for online vector balancing under arbitrary input distributions, and a $poly(\log T)$ bound for online interval discrepancy. Our framework is powerful enough to capture other well-studied geometric discrepancy problems; e.g., a $poly(\log^d (T))$ bound for the online $d$-dimensional Tusnády's problem. A key new technical ingredient is an {anti-concentration} inequality for sums of pairwise uncorrelated random variables.

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。

摘要：

英文摘要：

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