Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails (1404.1448v2)

Abstract: The Frechet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Frechet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n^2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Frechet distance cannot be computed in strongly subquadratic time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. Our result holds for both the continuous and the discrete Frechet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves).

• The paper establishes that under SETH, no strongly subquadratic algorithm exists for both continuous and discrete Fréchet distance computations.
• It proves that even a 1.001-approximation for the Fréchet distance cannot be achieved in strongly subquadratic time, barring breakthroughs in complexity theory.
• The work highlights imbalanced curve sizes and higher-dimensional extensions, emphasizing the practical and theoretical implications of these conditional lower bounds.

Analysis of "Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails"

The paper by Karl Bringmann addresses a longstanding question in computational geometry, specifically related to the computational complexity of calculating the Fréchet distance between two polygonal curves. Despite the extensive research and minor improvements over the years, the pursuit of an efficient algorithm for computing the Fréchet distance in under quadratic time has remained unresolved. Bringmann's work utilizes the Strong Exponential Time Hypothesis (SETH) to establish conditional lower bounds, effectively demonstrating that the Fréchet distance cannot be computed in strongly subquadratic time without conflicting with SETH.

Key Contributions

1. Conditional Lower Bound for Fréchet Distance: The principal contribution is the assertion that, under SETH, no strongly subquadratic algorithm exists for the computation of both continuous and discrete Fréchet distances. This suggests that attempts to find significantly faster algorithms for the Fréchet distance may not be feasible under currently accepted complexity-theoretic assumptions.
2. Approximation Hardness: Bringmann extends the analysis to approximation algorithms, arguing that even achieving a strongly subquadratic 1.001-approximation for the Fréchet distance is unlikely unless SETH fails. This closes the door to potential workaround solutions that leverage approximation.
3. Imbalance in Curve Sizes: The paper examines the scenario where the polygonal curves have significantly imbalanced numbers of vertices, offering insights into the computational complexity for this specific situation. A lower bound is established for any polynomial balancing of vertices between the two curves.
4. Realistic Input Models: Acknowledging that certain practical applications might involve specific types of inputs, the paper investigates the complexity of computing the Fréchet distance for $c$-packed curves. Bringmann shows that the current best approaches with respect to dependency on the parameter $c$ are likely optimal.
5. Extensions to Higher Dimensions: The paper outlines how the theoretic bounds extend into higher dimensions, particularly focusing on the field of $d$-dimensional space ($d \geq 5$), opening possibilities for further exploration in multidimensional data analysis and pattern recognition applications.

Implications and Future Directions

Bringmann's findings present formidable implications for both theoretical computer science and practical applications. The assumption of SETH renders the possibility of a breakthrough strongly subquadratic algorithm for the Fréchet distance rather unlikely using current paradigms.

For computational geometry researchers, these results recalibrate the focus towards either proving or refuting SETH or exploring alternative computational models. For practitioners relying on Fréchet distance computations, it suggests that current algorithmic approaches are possibly optimal in terms of core complexity, emphasizing instead the importance of heuristic or domain-specific optimizations.

Conclusion

The paper methodically addresses the complexity of computing the Fréchet distance by leveraging computational complexity theories, notably SETH, to establish a foundational understanding of its computational limitations. By doing so, it provides a robust framework that guides both future research and practical application developments, reinforcing the notion that fundamental complexity barriers shape the feasibility of faster algorithmic solutions.