Fr'echet Distance in Unweighted Planar Graphs

2025-04-24

AI Summary

This work investigates the computational complexity and efficient algorithms for computing the discrete Fréchet distance between two disjoint simple paths in unweighted planar graphs. Addressing an open question on the existence of a strongly subquadratic-time 1.25-approximation algorithm, we establish its impossibility under the Orthogonal Vectors Hypothesis (OVH), thereby tightening the known approximation lower bound. We then design an exact algorithm with running time Õ((|P| + |Q|)^1.5) for unit-weight regular grid graphs, achieving subquadratic performance. Furthermore, we introduce the notion of “well-structured curves”, enabling a (1+ε)-approximation algorithm. Our approach integrates graph-theoretic modeling, conditional complexity analysis, divide-and-conquer dynamic programming, and structural optimization tailored to grid-like topologies. Collectively, these contributions advance both the theoretical understanding and practical efficiency of trajectory similarity measurement on graphs.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

The Fr'echet distance is a distance measure between trajectories in the plane or walks in a graph G. Given constant-time shortest path queries in a graph G, the Discrete Fr'echet distance $F_G(P, Q)$ between two walks P and Q can be computed in $O(|P| cdot |Q|)$ time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly subquadratic algorithm to compute a $1.01$-approximation of $F_G(P, Q)$ unless the Orthogonal Vector Hypothesis (OVH) fails. Such quadratic-time conditional lower bounds are common to many Fr'echet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist $(1+varepsilon)$-approximations, both in weighted graphs and in Rd, that take near-linear time for $c$-packed or $kappa$-straight walks in the graph. In Rd, there also exists a near-linear time algorithm to compute the Fr'echet distance whenever all input edges are long compared to the distance. We consider computing the Fr'echet distance in unweighted planar graphs. We show that there exist no 1.25-approximations of the discrete Fr'echet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an $ ilde{O}( (|P| + |Q|)^{1.5})$-time algorithm to compute $D_F(P, Q)$. Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate $(1+varepsilon)$-approximations of their discrete Fr'echet distance in subquadratic time.

Problem

• Computing Fréchet distance in unweighted planar graphs efficiently
• Lower bounds for approximations in subquadratic time under OVH
• New well-behaved curve class enabling subquadratic approximations

Innovation

• Dynamic programming for Fréchet distance computation
• Subquadratic algorithm for regular tiling graphs
• New well-behaved curves class for approximations