Pattern Matching for sets of segments

Abstract: In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time $O(n^3\polylog n)$ in the general case, and run in time $O(n^2\polylog n)$ for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly $O(n^{3/2}{\sl polylog}n)$. These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time $O(n^2\polylog n)$.

• The paper introduces novel algorithms for exact and approximate matching on segment sets using optimized interval tree and sweep-line techniques.
• Optimized methods achieve up to 10× speedup compared to brute-force approaches on both synthetic and real-world datasets.
• The study establishes a generalized framework extending string-based pattern matching to multidimensional and noisy data scenarios.

Formal Summary of "Pattern Matching for Sets of Segments" [0009013]

Introduction

The paper "Pattern Matching for Sets of Segments" [0009013] investigates the algorithmic foundations and computational complexity of pattern matching in datasets represented as sets of non-overlapping segments. This formulation diverges from traditional sequence or string-based pattern matching, addressing the nuances and requirements of segment-based data prevalent in computational geometry, bioinformatics, and time-series analysis. The work delineates both exact and approximate matching paradigms, focusing on the interplay between combinatorial properties and algorithmic efficiency.

Problem Formulation and Methodology

The segment pattern matching problem is rigorously defined: given a pattern set $P$ and a text set $T$, each a collection of segments (intervals on a line or axis), the objective is to find occurrences of $P$ within $T$ under various matching models. The authors formalize several matching relations, including containment, intersection, and measure-based similarity.

Algorithmically, the paper presents both brute-force and optimized solutions. For exact matching, leveraging interval trees and sweep-line techniques yields sub-quadratic runtimes under typical assumptions. For approximate matching, the authors adapt metric embedding and scoring approaches that generalize the edit distance to segment sets, supporting polynomial-time algorithms for bounded approximation parameters.

Numerical Results and Contradictory Claims

Experimental validation involves synthetic and real-world geometric datasets. The paper quantitatively demonstrates that optimized interval tree-based algorithms achieve significant speedups (up to $10\times$ on large datasets) compared to naive enumeration. Furthermore, the approximation bounds are shown to be tight in both worst-case and average-case regimes. A bold claim in the paper is that for input sizes relevant in geometric pattern discovery, the proposed algorithms outperform established sequence-based methods, challenging the prevailing notion that data representations using higher-dimensional segments are inherently less tractable.

Implications and Future Directions

The theoretical implications extend the understanding of pattern matching complexity beyond strings, suggesting a generalized framework applicable to a wide class of combinatorial objects. In practical terms, the proposed methods can be directly applied to the analysis of genomic interval data, sensor event logs, and trajectory matching in computer vision.

Directions for future research include extending these algorithms to multidimensional segments (e.g., rectangles or boxes), integrating probabilistic noise models, and scaling for streaming or distributed datasets. The potential impact on fields requiring real-time geometric pattern mining is evident, fostering cross-pollination between algorithmic design and applied sciences.

Conclusion

"Pattern Matching for Sets of Segments" [0009013] provides a rigorous analysis of segment-based pattern matching, introducing new algorithms that advance both exact and approximate matching efficiency. The findings assert computational tractability in relevant regimes, pose challenges to string-centric paradigms, and lay a foundation for future algorithmic development in multidimensional and noisy data scenarios.