高维一致数字射线和二维部分一致数字射线的距离界
Distance bounds for high dimensional consistent digital rays and 2-D partially-consistent digital rays
摘要：
我们考虑数字化欧几里德段的问题。具体来说，我们寻找一个建设性的方法来连接任何两个点在$\mathbb{Z}^d$。构造必须是{ \em consistent}（即满足欧几里得公理的自然扩展），同时尽可能地与它们相似。以前的工作已经显示出渐近紧的结果，在两个维度与$Θ(\log N)$错误，其中段之间的相似性测量与豪斯多夫距离，和$N$是$L_1$距离之间的两个点。这种构造被认为是紧的，因为$Ω(\log N)$下限适用于$\mathbb{Z}^2$中的任何一致构造。在本文中，我们观察到，下限并不直接扩展到高维。我们给出了另一种论证，证明了$d$维数中的任何相容构造都必须有$Ω(\log^{1/(d-1)} N)$误差。我们将高维一致构造的误差与二维类似的{ \em weak}构造的误差联系起来（某些点不需要满足所有公理的构造）。这不仅打开了在高维中具有$o(\log N)$错误的构造的可能性，而且在公理违反的数量和构造的错误之间的权衡中开辟了一条有趣的研究路线。为了显示我们的下限，我们还考虑了一个彩色的变化的概念，我们发现一组点的差异的独立利益。
英文摘要：
We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with $Θ(\log N)$ error, where resemblance between segments is measured with the Hausdorff distance, and $N$ is the $L_1$ distance between the two points. This construction was considered tight because of a $Ω(\log N)$ lower bound that applies to any consistent construction in $\mathbb{Z}^2$. In this paper we observe that the lower bound does not directly extend to higher dimensions. We give an alternative argument showing that any consistent construction in $d$ dimensions must have $Ω(\log^{1/(d-1)} N)$ error. We tie the error of a consistent construction in high dimensions to the error of similar {\em weak} constructions in two dimensions (constructions for which some points need not satisfy all the axioms). This not only opens the possibility for having constructions with $o(\log N)$ error in high dimensions, but also opens up an interesting line of research in the tradeoff between the number of axiom violations and the error of the construction. In order to show our lower bound, we also consider a colored variation of the concept of discrepancy of a set of points that we find of independent interest.
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