Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic
| Autor: | Arnold Filtser, Omrit Filtser, Matthew J. Katz |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Computational Geometry (cs.CG)
FOS: Computer and information sciences General Computer Science Applied Mathematics Theory of computation → Design and analysis of algorithms Theory of computation → Computational geometry polygonal curves Computer Science Applications Fréchet distance range counting dynamic time warping Computer Science - Computational Geometry (asymmetric) approximate nearest neighbor approximation algorithms |
| DOI: | 10.4230/lipics.icalp.2020.48 |
| Popis: | In the $(1+\varepsilon,r)$-approximate near-neighbor problem for curves (ANNC) under some distance measure $\delta$, the goal is to construct a data structure for a given set $\mathcal{C}$ of curves that supports approximate near-neighbor queries: Given a query curve $Q$, if there exists a curve $C\in\mathcal{C}$ such that $\delta(Q,C)\le r$, then return a curve $C'\in\mathcal{C}$ with $\delta(Q,C')\le(1+\varepsilon)r$. There exists an efficient reduction from the $(1+\varepsilon)$-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve $C\in\mathcal{C}$ with $\delta(Q,C)\le(1+\varepsilon)\cdot\delta(Q,C^*)$, where $C^*$ is the curve of $\mathcal{C}$ closest to $Q$. Given a set $\mathcal{C}$ of $n$ curves, each consisting of $m$ points in $d$ dimensions, we construct a data structure for ANNC that uses $n\cdot O(\frac{1}{\varepsilon})^{md}$ storage space and has $O(md)$ query time (for a query curve of length $m$), where the similarity between two curves is their discrete Fr\'echet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds. Further, we also consider the asymmetric version of ANNC, where the length of the query curves is $k \ll m$, and obtain essentially the same storage and query bounds as above, except that $m$ is replaced by $k$. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds. |
| Databáze: | OpenAIRE |
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