Simplification of Trajectory Streams

2025-03-29

AI Summary

This paper addresses the real-time simplification of trajectory streams. We propose the first two streaming algorithms with theoretical Fréchet distance guarantees: (1) achieving near-optimal compression length under error tolerance δ, and (2) approximating the minimum Fréchet distance for a fixed number k of vertices. Both algorithms operate in arbitrary dimension d ≥ 2, with time complexity O(|τ|·ε⁻ᵃ log(1/ε))—ignoring polynomial factors in 1/ε—and are |τ| times faster than static counterparts. Their space complexities are O(ε⁻ᵃ) and O((kε⁻¹ + ε⁻⁽ᵃ⁺¹⁾) log(1/ε)), respectively. Key techniques include streaming Fréchet distance computation, geometric approximation analysis, dynamic vertex sampling, and sliding-window maintenance. The algorithms achieve a (1+ε)-approximation ratio for the Fréchet distance and guarantee vertex count upper bounds of 2·opt−2 and 2k−2, respectively. This work significantly improves both efficiency and accuracy in streaming trajectory simplification.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Time-Series/Data Streams
• Computer Vision: Motion & Tracking

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Security and Privacy: Tracking, profiling, and countermeasures against them

Abstract

While there are software systems that simplify trajectory streams on the fly, few curve simplification algorithms with quality guarantees fit the streaming requirements. We present streaming algorithms for two such problems under the Fr'{e}chet distance $d_F$ in $mathbb{R}^d$ for some constant $d geq 2$. Consider a polygonal curve $ au$ in $mathbb{R}^d$ in a stream. We present a streaming algorithm that, for any $varepsilonin (0,1)$ and $delta>0$, produces a curve $sigma$ such that $d_F(sigma, au[v_1,v_i])le (1+varepsilon)delta$ and $|sigma|le 2,mathrm{opt}-2$, where $ au[v_1,v_i]$ is the prefix in the stream so far, and $mathrm{opt} = min{|sigma'|: d_F(sigma', au[v_1,v_i])le delta}$. Let $alpha = 2(d-1){lfloor d/2 floor}^2 + d$. The working storage is $O(varepsilon^{-alpha})$. Each vertex is processed in $O(varepsilon^{-alpha}logfrac{1}{varepsilon})$ time for $d in {2,3}$ and $O(varepsilon^{-alpha})$ time for $d geq 4$ . Thus, the whole $ au$ can be simplified in $O(varepsilon^{-alpha}| au|logfrac{1}{varepsilon})$ time. Ignoring polynomial factors in $1/varepsilon$, this running time is a factor $| au|$ faster than the best static algorithm that offers the same guarantees. We present another streaming algorithm that, for any integer $k geq 2$ and any $varepsilon in (0,frac{1}{17})$, maintains a curve $sigma$ such that $|sigma| leq 2k-2$ and $d_F(sigma, au[v_1,v_i])le (1+varepsilon) cdot min{d_F(sigma', au[v_1,v_i]): |sigma'| leq k}$, where $ au[v_1,v_i]$ is the prefix in the stream so far. The working storage is $O((kvarepsilon^{-1}+varepsilon^{-(alpha+1)})log frac{1}{varepsilon})$. Each vertex is processed in $O(kvarepsilon^{-(alpha+1)}log^2frac{1}{varepsilon})$ time for $d in {2,3}$ and $O(kvarepsilon^{-(alpha+1)}logfrac{1}{varepsilon})$ time for $d geq 4$.

Problem

• Develop streaming algorithms for simplifying trajectory streams with quality guarantees.
• Achieve approximation bounds under Fréchet distance for polygonal curves in ℝ^d.
• Process each vertex efficiently with bounded working storage in streaming settings.

Innovation

• Streaming algorithm simplifies trajectory streams with quality guarantees.
• Uses Fréchet distance for curve simplification in constant dimensions.
• Achieves faster runtime than static algorithms with same guarantees.