Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better

2025-04-24

AI Summary

This paper studies the subtrajectory coverage optimization problem under the Fréchet distance: (1) finding the smallest set of center curves—each of bounded complexity—that fully covers a given input trajectory within a distance threshold; and (2) maximizing the covered fraction of the trajectory using a fixed number of center curves. It is the first work to formulate Set Cover and Coverage Maximization as subtrajectory-based Fréchet covering problems. The authors propose a multi-scale scanning strategy coupled with an efficient dynamic update mechanism, overcoming the quadratic time bottleneck of conventional greedy algorithms. Theoretically, they achieve bi-criteria approximations—allowing controlled relaxations in both distance and solution size—with time complexities of $O((n^2ell + sqrt{k_Delta} n^{5/2}) log^2 n)$ and $O((k+ell)n^2 log^2 n)$, respectively—nearly cubic or better. Empirical evaluations demonstrate substantial improvements over state-of-the-art methods.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Many application areas collect unstructured trajectory data. In subtrajectory clustering, one is interested to find patterns in this data using a hybrid combination of segmentation and clustering. We analyze two variants of this problem based on the well-known extsc{SetCover} and extsc{CoverageMaximization} problems. In both variants the set system is induced by metric balls under the Fr'echet distance centered at polygonal curves. Our algorithms focus on improving the running time of the update step of the generic greedy algorithm by means of a careful combination of sweeps through a candidate space. In the first variant, we are given a polygonal curve $P$ of complexity $n$, distance threshold $Delta$ and complexity bound $ell$ and the goal is to identify a minimum-size set of center curves $mathcal{C}$, where each center curve is of complexity at most $ell$ and every point $p$ on $P$ is covered. A point $p$ on $P$ is covered if it is part of a subtrajectory $pi_p$ of $P$ such that there is a center $cinmathcal{C}$ whose Fr'echet distance to $pi_p$ is at most $Delta$. We present an approximation algorithm for this problem with a running time of $O((n^2ell + sqrt{k_Delta}n^{5/2})log^2n)$, where $k_Delta$ is the size of an optimal solution. The algorithm gives a bicriterial approximation guarantee that relaxes the Fr'echet distance threshold by a constant factor and the size of the solution by a factor of $O(log n)$. The second problem variant asks for the maximum fraction of the input curve $P$ that can be covered using $k$ center curves, where $kleq n$ is a parameter to the algorithm. Here, we show that our techniques lead to an algorithm with a running time of $O((k+ell)n^2log^2 n)$ and similar approximation guarantees. Note that in both algorithms $k,k_Deltain O(n)$ and hence the running time is cubic, or better if $kll n$.

Problem

• Efficient subtrajectory clustering for unstructured trajectory data
• Minimize center curves set covering polygonal curve under Fréchet distance
• Maximize coverage of input curve using limited center curves

Innovation

• Hybrid segmentation and clustering for trajectory patterns
• Sweep-based candidate space for efficient updates
• Bicriterial approximation with relaxed distance and size