Property Testing of Curve Similarity

2025-02-24

AI Summary

This work addresses the sublinear-query problem of curve similarity testing: given two curves, determine with minimal queries whether their discrete or continuous Fréchet distance is ≤ δ, or whether more than an ε-fraction of points must be ignored to achieve δ-similarity. It introduces the property testing paradigm to curve similarity verification for the first time. The authors propose a prior-free adaptive query strategy and extend it to (1+ε′)-approximate testing of the continuous Fréchet distance. Their structured analytical framework combines oracle-based sampling, matrix modeling, and the t-approximate shortest path assumption. Under the t-straightness assumption—where each curve admits a t-piecewise-linear approximation—the query complexity is O(t/ε log(t/ε)) for the discrete case and O((t³ + t² log n)/ε) for the continuous case. These bounds substantially improve upon full-distance computation and support high-dimensional and streaming settings.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

We propose sublinear algorithms for probabilistic testing of the discrete and continuous Fr'echet distance - a standard similarity measure for curves. We assume the algorithm is given access to the input curves via a query oracle: a query returns the set of vertices of the curve that lie within a radius $delta$ of a specified vertex of the other curve. The goal is to use a small number of queries to determine with constant probability whether the two curves are similar (i.e., their discrete Fr'echet distance is at most $delta$) or they are ''$varepsilon$-far'' (for $0<varepsilon<2$) from being similar, i.e., more than an $varepsilon$-fraction of the two curves must be ignored for them to become similar. We present two algorithms which are sublinear assuming that the curves are $t$-approximate shortest paths in the ambient metric space, for some $tll n$. The first algorithm uses $O(frac{t}{varepsilon}logfrac{t}{varepsilon})$ queries and is given the value of $t$ in advance. The second algorithm does not have explicit knowledge of the value of $t$ and therefore needs to gain implicit knowledge of the straightness of the input curves through its queries. We show that the discrete Fr'echet distance can still be tested using roughly $O(frac{t^3+t^2log n}{varepsilon})$ queries ignoring logarithmic factors in $t$. Our algorithms work in a matrix representation of the input and may be of independent interest to matrix testing. Our algorithms use a mild uniform sampling condition that constrains the edge lengths of the curves, similar to a polynomially bounded aspect ratio. Applied to testing the continuous Fr'echet distance of $t$-straight curves, our algorithms can be used for $(1+varepsilon')$-approximate testing using essentially the same bounds as stated above with an additional factor of poly$(frac{1}{varepsilon'})$.

Problem

• Sublinear algorithms for curve similarity testing
• Probabilistic testing of Fréchet distance
• Query-efficient curve similarity determination

Innovation

• Sublinear algorithms for curve similarity
• Query oracle for vertex access
• Matrix representation for curve testing