曲线的局部敏感散列
Locality-sensitive hashing of curves
摘要：
我们研究的数据结构存储一组多边形曲线在${\rm R}^d$，这样，给定一个查询曲线，我们可以有效地检索相似的曲线从集合中，其中相似性是使用离散的Fréchet距离或动态时间弯曲距离。为此，我们设计了第一个位置敏感的哈希方案，这些距离的措施。一个主要的挑战是，这些距离测量内部优化曲线之间的对齐。我们给出了不同类型的路线，包括约束和无约束的版本的解决方案。对于无约束的路线，我们改进了Indyk从2002年开始的短曲线的结果。令$n$为输入曲线的数量，并令$m$为输入中曲线的最大复杂度。在特定的情况下，$m \leq \fracα{4d} \log n$，对于一些固定的$α>0$，我们的解决方案意味着一个近似的近邻数据结构的离散弗雷歇距离，使用空间在$O(n^{1+α}\log n)$和实现查询时间在$O(n^α\log^2 n)$和常数近似因子。此外，我们的解决方案提供了近似质量和计算性能之间的权衡：对于任何参数$k \in [m]$，我们可以给出一个数据结构，使用空间在$O(2^{2k}m^{k-1} n \log n + nm)$，回答查询在$O( 2^{2k} m^{k}\log n)$时间，并实现近似因子在$O(m/k)$。
英文摘要：
We study data structures for storing a set of polygonal curves in ${\rm R}^d$ such that, given a query curve, we can efficiently retrieve similar curves from the set, where similarity is measured using the discrete Fréchet distance or the dynamic time warping distance. To this end we devise the first locality-sensitive hashing schemes for these distance measures. A major challenge is posed by the fact that these distance measures internally optimize the alignment between the curves. We give solutions for different types of alignments including constrained and unconstrained versions. For unconstrained alignments, we improve over a result by Indyk from 2002 for short curves. Let $n$ be the number of input curves and let $m$ be the maximum complexity of a curve in the input. In the particular case where $m \leq \fracα{4d} \log n$, for some fixed $α>0$, our solutions imply an approximate near-neighbor data structure for the discrete Fréchet distance that uses space in $O(n^{1+α}\log n)$ and achieves query time in $O(n^α\log^2 n)$ and constant approximation factor. Furthermore, our solutions provide a trade-off between approximation quality and computational performance: for any parameter $k \in [m]$, we can give a data structure that uses space in $O(2^{2k}m^{k-1} n \log n + nm)$, answers queries in $O( 2^{2k} m^{k}\log n)$ time and achieves approximation factor in $O(m/k)$.
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