The fundamental model of a periodic structure is a periodic set of points considered up to rigid motion or isometry in Euclidean space. The recent work by Edelsbrunner et al defined the new isometry invariants (density functions), which are continuous under perturbations of points and complete for generic sets in dimension 3. This work introduces much faster invariants called higher order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented by numerical matrices and are also continuous under perturbations important for applications. Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also confirmed by distinguishing all 229K known molecular organic structures from the world's largest Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop due to the proposed algorithm with a near linear or small polynomial complexity in terms of key input sizes. Most importantly, the above completeness allows one to build a common map of all periodic structures, which are continuously parameterized by PDD and explicitly reconstructible from PDD. Appendices include first tree-based maps for several thousands of real structures.

翻译：周期结构的基本模型是周期性结构的基本模型,这是一套定期的点数,可以考虑在Euclidean空间进行僵硬运动或同位素测量。Edelsbrunner等人最近的工作确定了新的异变体(密度功能),这些新异变体在点的扰动下连续进行,并完整完成3级的通用结构。这种工作引入了更快速的异变体,称为PDDP(PDD),以数字矩阵为代表,新的PDD(DDD)变异体也在对应用很重要的扰动中连续进行。PDD(DD)的完整性被证明为任何层面的远距离基因组,这一点也得到确认,将所有229K已知的分子有机结构与世界上最大的剑桥结构数据库区分开来加以确认。这一巨大的实验仅仅用了7小时,因为拟议的算法在关键输入大小方面几乎线性或小的多元性复合性。最重要的是,上述完整性允许人们建立所有定期结构的共同地图,这些结构由PDDD(PDD)持续参数和从PDDDD(PDDD)明确可重建。Appendices)的最初以树为基础的地图。