[论文解读] Efficient Neighbor-Finding on Space-Filling Curves

David Holzmüller|arXiv (Cornell University)|Oct 17, 2017

Computational Geometry and Mesh Generation参考文献 5被引用 4

一句话总结

本文提出了一种在空间填充曲线上进行邻近查找的高效算法，利用在矩阵等价类上的半群结构中的转移矩阵。通过在商空间上定义右群作用，该方法实现了具有正确性和性能理论保证的快速空间邻近查询，在多维索引中显著优于朴素方法。

ABSTRACT

Space-filling curves (SFC, also known as FASS-curves) are a useful tool in scientific computing and other areas of computer science to sequentialize multidimensional grids in a cache-efficient and parallelization-friendly way for storage in an array. Many algorithms, for example grid-based numerical PDE solvers, have to access all neighbor cells of each grid cell during a grid traversal. While the array indices of neighbors can be stored in a cell, they still have to be computed for initialization or when the grid is adaptively refined. A fast neighbor-finding algorithm can thus significantly improve the runtime of computations on multidimensional grids. In this thesis, we show how neighbors on many regular grids ordered by space-filling curves can be found in an average-case time complexity of $O(1)$. In general, this assumes that the local orientation (i.e. a variable of a describing grammar) of the SFC inside the grid cell is known in advance, which can be efficiently realized during traversals. Supported SFCs include Hilbert, Peano and Sierpinski curves in arbitrary dimensions. We assume that integer arithmetic operations can be performed in $O(1)$, i.e. independent of the size of the integer. We do not deal with the case of adaptively refined grids here. However, it appears that a generalization of the algorithm to suitable adaptive grids is possible. To formulate the neighbor-finding algorithm and prove its correctness and runtime properties, a modeling framework is introduced. This framework extends the idea of vertex-labeling to a description using grammars and matrices. With the sfcpp library, we provide a C++ implementation to render SFCs generated by such models and automatically compute all lookup tables needed for the neighbor-finding algorithm. Furthermore, optimized neighbor-finding implementations for various SFCs are included for which we provide runtime measurements.

研究动机与目标

• 解决在高维空间中空间填充曲线上邻近查找的计算低效问题。
• 将转移矩阵的结构形式化为在矩阵等价类上作用的半群。
• 通过在商空间上利用代数群作用，实现快速且正确的邻近查询。
• 提供一个数学上严谨的框架，支持可扩展的多维空间索引。

提出的方法

• 本文在 ℝ^{d×m} 上定义了一个等价关系 ∼，并考虑商空间 ℝ^{d×m}/{∼}。
• 引入 ℝ^{m×n} 中的转移矩阵集合 ℳ，其在矩阵运算下构成一个半群。
• 通过矩阵乘法定义 ℳ 在商空间 [Q] 上的右群作用：[Q]⋅M。
• 该作用被构造为保持空间局部性，确保曲线中的邻近点对应于相近的索引。
• 该方法利用半群的代数性质，将邻近查找简化为矩阵运算。
• 该框架通过利用转移矩阵的结构，支持邻近点的高效计算。

实验结果

研究问题

• RQ1 如何利用代数结构对空间填充曲线上邻近查找进行形式化？
• RQ2 转移矩阵的哪些代数性质能够支持高效的邻近查询？
• RQ3 在商空间上的群作用能否在多维索引中保持空间局部性？
• RQ4 转移矩阵的半群结构如何提升查询性能？

主要发现

• 在矩阵乘法下，转移矩阵集合 ℳ 构成一个半群，支持结构化计算。
• ℳ 在商空间 ℝ^{d×m}/{∼} 上的右群作用保持了邻近查找的空间局部性。
• 该框架通过将邻近查询简化为等价类上的矩阵运算，实现了高效的邻近查询。
• 该方法为高维空间中可扩展的空间索引提供了数学上严谨的基础。