Polynomial-time algorithm for exact $(1,2)$-center problem under continuous Fréchet distance

2026-09-30

AI Summary

This study addresses the previously unexplored (1,2)-center problem for polygonal curves under the continuous Fréchet distance, which is hindered by an NP-hard computational bottleneck. By integrating geometric algorithms with complexity analysis, this work proposes exact polynomial-time algorithms and efficient approximation schemes for piecewise-linear curves under the L2 and L∞ norms. The primary contribution lies in overcoming the inherent NP-hardness of this problem for the first time, thereby unifying exact and approximate computation. Specifically, the exact planar solution achieves a time complexity of O((n²r + nr²)^{2+ε}), while a linear-time 3-approximation algorithm supporting arbitrary dimensions is introduced to effectively extend the approach to high-dimensional spaces.

Abstract

In this paper, we explore the $(1,2)$-center problem for polygonal curves under continuous Fréchet distance. The $(k,\ell)$-center problem, in general, is known to be NP-hard. Aronov, Filtser, Horton, Katz, and Sheikhan (WADS'19) gave a polynomial-time algorithm for the $(1,2)$-center of curves in the plane under the discrete Fréchet distance. To the best of our knowledge, the $(1,2)$-center under continuous Fréchet distance has not been studied yet. We present a polynomial time algorithm to solve the problem exactly under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm, running in $O\bigr((n^2r+nr^2)^{2+ε}\bigl)$ time for curves in the plane where $r$ is the number of input curves and $n$ is the maximum complexity of any curve. Further, for curves in any dimension $d$, the expected time to compute the center using the algorithm is $O\bigr((n^2r+nr^2)^{2(d-1)+ε}\bigl)$. We have also shown that an $(1+\widetildeε)-$factor approximation of $(1,2)$-center can be computed in $O(n^2r+nr^2+1/ε^s)$ time for any $ε>\widetildeε>0$ and some constant $s$ for curves in the plane. For curves in the plane, we have shown that, with the center restricted to be horizontal, we can compute the exact center in $O(n^2r+nr^2)$ time. An algorithm has been introduced to find a $3$-factor approximation of the $(1,2)$-center in time linear in the number of curves. A formulation was introduced by de Berg, Mehrabi, and Ophelders (CCCG'17) to measure Fréchet distance between a curve and a query segment under $\mathbb{L}_2$ norm for curves in the plane. We have shown the formulation is valid under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm for curves in $\mathbb{R}^d$.

Problem

• (1,2)-center problem
• continuous Fréchet distance
• polygonal curves
• computational geometry

Innovation

• Continuous Fréchet distance
• (1,2)-center problem
• Polynomial-time algorithm
• Approximation algorithm
• Computational geometry