[Paper Review] On the Approximate Nearest Neighbor Queries among Curves under the Fréchet Distance

Majid Mirzanezhad|arXiv (Cornell University)|Apr 17, 2020

Computational Geometry and Mesh Generation35 references4 citations

TL;DR

This paper presents the first deterministic data structure for approximate near-neighbor search (ANNS) among polygonal curves under the continuous Fréchet distance in ℝᵈ. It achieves (1+ε)δ-approximate queries in O(kd) time with space complexity n·O((√d/ε)ᵏᵈ) when k is known, or O((1/ε)ᵐᵈ) space when k is part of the query, enabling efficient subtrajectory and time-window queries with constant-time response.

ABSTRACT

Approximate near-neighbors search ( extsc{ANNS}) is a long-studied problem in computational geometry. %that has received considerable attention by researchers in the community. In this paper, we revisit the problem and propose the first data structure for curves under the (continuous) Fréchet distance in $\Reals^d$. Given a set $¶$ of $n$ curves of size at most $m$ each in $\Reals^d$, and a real fixed $δ>0$, we aim to preprocess $¶$ into a data structure so that for any given query curve $Q$ of size $k$, we can efficiently report all curves in $¶$ whose Fréchet distances to $Q$ are at most $δ$. In the case that $k$ is given in the preprocessing stage, for any $\eps>0$ we propose a deterministic data structure whose space is $n \cdot O\big(\max\big\{\big(\frac{\sqrt{d}}{\eps}\big)^{kd}, \big(\frac{\D\sqrt{d}}{\eps^2}\big)^{kd}\big\}\big)$ that can answer extsc{$(1+\eps)δ$-ANNS} queries in $O(kd)$ query time, where $\D$ is the diameter of $¶$. Considering $k$ as part of the query slightly changes the space to $n \cdot O\big(\frac{1}{\eps}\big)^{md} $ with $O(kd)$ query time within an approximation factor of $5+\eps$. We show that our generic data structure for ANNS can give an alternative treatment of the approximate subtrajectory range searching problem studied by de Berg et al. [8]. We also revisit the time-window data structure for spatial density maps in [6]. Given $θ>0$, and $n$ time-stamped points spread over $m$ regions in a map, for any query window $W$, we propose a data structure of size $O(n/\eps^2)$ and construction time $O((n+m)/\eps^2)$ that can approximately return the regions containing at least $θ$ points whose times are within $W$ in $O(1)$ query time.

Motivation & Objective

• To address the long-standing challenge of approximate near-neighbor search (ANNS) among curves under the continuous Fréchet distance in high-dimensional space ℝᵈ.
• To design a deterministic, efficient data structure that supports (1+ε)δ-approximate queries for any query curve Q of size k.
• To extend the framework to handle related problems such as approximate subtrajectory range searching (ASRS) and time-window queries for spatial density maps.
• To provide a simplified and efficient alternative to prior methods, particularly for time-window data structures in spatiotemporal databases.

Proposed method

• Propose a data structure that discretizes the space of possible curve alignments using grid-based rounding of time and position parameters to enable efficient lookup.
• Use a hash table to index regions or curve segments based on rounded time-windows, enabling constant-time query resolution.
• Apply rounding techniques to transform query windows [q₁, q₂] into nearby grid-aligned windows W₁ and W₂ to bound approximation error.
• Leverage the property that S₁ ⊆ S* ⊆ S₂ for rounded windows W₁ ⊆ W₂, ensuring correctness of the (1+ε)-approximation.
• Design a space-efficient structure for ANNS by bounding the number of relevant curve alignments using (ε, d, k)-nets and grid quantization.
• Integrate the method into time-window data structures, achieving O(1) query time with O(n/ε²) space and O((n+m)/ε²) construction time.

Experimental results

Research questions

• RQ1 Can a deterministic data structure be constructed for ANNS among curves under the continuous Fréchet distance in ℝᵈ with sublinear space and efficient query time?
• RQ2 How can the approximation factor be controlled to (1+ε)δ while maintaining practical space and query complexity?
• RQ3 Can the proposed method be extended to support approximate subtrajectory range searching (ASRS) and time-window queries in spatiotemporal databases?
• RQ4 Is it possible to achieve constant-time query response for time-window queries with linear-size data structures and sublinear preprocessing?

Key findings

• The proposed data structure supports (1+ε)δ-approximate ANNS queries in O(kd) time with space complexity n·O((√d/ε)ᵏᵈ) when k is known at preprocessing time.
• When k is not fixed at preprocessing, the space increases to n·O((1/ε)ᵐᵈ), but query time remains O(kd), with an approximation factor of 5+ε.
• The method achieves O(1) query time for time-window queries with a data structure of size O(n/ε²) and construction time O((n+m)/ε²).
• The framework provides a (1+ε)-approximation for time-window queries by using two grid-rounded windows W₁ and W₂, ensuring S₁ ⊆ S* ⊆ S₂.
• The approach offers a simplified and deterministic alternative to prior randomized or complex methods in ANNS and ASRS under Fréchet distance.
• The method can be adapted to handle both continuous and discrete Fréchet distances, and supports asymmetric and symmetric cases.