Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

Abstract: We present simple and practical $(1+\eps)$-approximation algorithm for the Frechet distance between curves. To analyze this algorithm we introduce a new realistic family of curves, $c$-packed curves, that is closed under simplification. We believe the notion of $c$-packed curves to be of independent interest. We show that our algorithm has near linear running time for $c$-packed curves, and show similar results for other input models.

• The paper introduces a (1+ε)-approximation algorithm for the Fréchet distance between polygonal curves using a novel realistic input model called c-packed curves.
• The algorithm achieves near linear time complexity, making it efficient for practical applications and large-scale datasets.
• This approach offers significant theoretical contributions, including the introduction of c-packed curves, and has practical implications in areas like shape comparison for GIS, molecular biology, and CAD.

Approximating the $e$ Distance for Realistic Curves in Near Linear Time

This paper introduces a computational method to approximate the $e$ (Fréchet) distance between polygonal curves, an essential metric in computational geometry used for shape comparison. The study is situated in the field of $d$-dimensional space and provides a practical algorithm with near linear running time, contingent upon specific modeling assumptions.

Overview

The authors present a simple and effective $(1+\eps)$-approximation algorithm to calculate the $e$ distance between two polygonal curves, leveraging the notion of $c$-packed curves. These curves, characterized by limited total length within any given ball, serve as a realistic input model, enabling efficient computation. The algorithm's running time is near linear for $c$-packed curves, and similar efficiency is noted for other input models such as low-density polygonal curves.

Algorithmic Specifics

The algorithm employs a curve simplification technique, which allows for a significant reduction in computational complexity by decreasing the free space diagram's size, a pivotal element in calculating the $e$ distance. The simplification adheres to the intrinsic structural constraints of the curves without loss of essential distance metrics.

The methodology involves:

• Simplifying input curves based on a parameter $\eps$, yielding a simplified curve that retains meaningful geometric properties.
• Introducing a mechanism to compute a reachable free space diagram, effectively capturing feasible paths representative of the actual $e$ distance.
• Conducting a series of binary searches within defined intervals informed by event radii, which encapsulate changes in the free space diagram structure due to vertex-edge and monotonicity events.

Theoretical Contributions

The study's theoretical implications are substantial. The introduction of $c$-packed curves is a central contribution, providing a rigorous yet practically applicable family of curves that encapsulate realistic behavior under varying resolutions. These curves have notable qualities:

• Independence from space dimension, yielding broad applicability across various geometric configurations.
• Closure under simplification, ensuring stable properties when subjected to approximation techniques.

Further theoretical extensions include:

• Establishing bounds for the relative free space complexity of low-density curves and $\kappa$-bounded curves, offering improvements over prior results and extending the approximation algorithm's applicability.
• Adapting the algorithm for closed curves, adding versatility to the approach and addressing additional complexities inherent in closed loop structures.

Implications and Future Work

The implications of this work extend beyond theoretical development, offering potential practical applications across various domains necessitating shape comparison, such as geographic information systems, molecular biology, and computer-aided design. The algorithm's simplicity and efficiency present a feasible tool for practitioners working with large-scale datasets.

Prospective future research directions could explore:

• Applications in dynamic environments where curves evolve over time, requiring real-time or adaptive approximation methods.
• Further refinement of simplification techniques to enhance computational efficiency without compromising accuracy.
• Expanding the framework to handle more complex geometric objects beyond polygonal curves.

In conclusion, this work bridges theoretical rigor with practical computational efficiency, setting a foundation for further explorations in shape comparison using Fréchet distance approximations. By introducing and leveraging the $c$-packed curve model, the study offers a substantial improvement in computational methods for analyzing continuous and structured geometric forms.