连续 Fréchet 距离下精确 $(1,2)$-中心问题的多项式时间算法

中文摘要 AI 辅助

本文研究了在连续 Fréchet 距离下多边形曲线的 $(1,2)$-中心问题。一般而言，$(k,\ell)$-中心问题已知是 NP 困难的。Aronov、Filtser、Horton、Katz 和 Sheikhan（WADS'19）给出了在离散 Fréchet 距离下平面曲线 $(1,2)$-中心的多项式时间算法。据我们所知，在连续 Fréchet 距离下的 $(1,2)$-中心问题尚未被研究。我们提出了一种多项式时间算法，在 $\mathbb{L}_2$ 和 $\mathbb{L}_\infty$ 范数下精确求解该问题，对于平面曲线，其运行时间为 $O\bigr((n^2r+nr^2)^{2+\epsilon}\bigl)$，其中 $r$ 是输入曲线的数量，$n$ 是任意曲线的最大复杂度。进一步，对于任意维度 $d$ 中的曲线，使用该算法计算中心的期望时间为 $O\bigr((n^2r+nr^2)^{2(d-1)+\epsilon}\bigl)$。我们还证明了对于平面曲线，对于任意 $\epsilon>\widetilde{\epsilon}>0$ 和某个常数 $s$，可以在 $O(n^2r+nr^2+1/\epsilon^s)$ 时间内计算 $(1,2)$-中心的 $(1+\widetilde{\epsilon})-$ 因子近似。对于平面曲线，我们证明了当中心被限制为水平时，可以在 $O(n^2r+nr^2)$ 时间内计算精确中心。我们引入了一种算法，在曲线数量线性时间内找到 $(1,2)$-中心的 $3$-因子近似。de Berg、Mehrabi 和 Ophelders（CCCG'17）提出了一种公式，用于在 $\mathbb{L}_2$ 范数下测量平面曲线与查询线段之间的 Fréchet 距离。我们证明了该公式在 $\mathbb{R}^d$ 中的曲线在 $\mathbb{L}_2$ 和 $\mathbb{L}_\infty$ 范数下均有效。

英文摘要

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 $\mathbb{R}^d$.

发表机构

• Indian Statistical Institute（印度统计研究所）
• Indian Institute of Technology Delhi（德里印度理工学院）

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