一种降维的Fréchet简化神谕

中文摘要 AI 辅助

设P为平面上具有n个顶点的多边形曲线。我们构建了一个大小为O(n log n)的数据结构，适用于以下类型的简化查询：给定一条查询直线ℓ和一个整数k≥1，在所有此类曲线中，找到直线ℓ上顶点数不超过k的曲线Q，使其与P的离散Fréchet距离最小。利用我们的数据结构，查询可在O(k² log³n + k log⁴n)时间内处理。更一般地，平面上具有n个顶点的几何树T可被预处理为近线性大小的结构，使得给定其两个顶点u、v、一条直线ℓ和一个整数k≥1，可在O(k² polylog n)时间内找到直线ℓ上顶点数不超过k的曲线Q，使其与T中从u到v的路径的离散Fréchet距离最小。对于一般降维问题，当P为ℝᵈ（d≥3）中的曲线、0<ε₀<1为实参数、查询指定一个g-平坦h（1≤g≤d-1）和整数k≥1时，我们构建了一个大小为O(n log n + f(ε₀)n)的数据结构，其中f(ε₀)=(1+1/ε₀)^((d-1)/2)，该结构允许我们找到h上顶点数不超过k的曲线Q，其与P的离散Fréchet距离最多为Q*与P距离的1+ε₀倍，其中Q*是使与P距离最小的此类曲线，查询处理时间为O(f(ε₀)k² log²n)。

英文摘要

Let $P$ be a polygonal curve with $n$ vertices in the plane. We construct a data structure of size $O(n \log n)$ suited for simplification queries of the following kind. Given a query line $\ell$ and an integer $k\ge1$, find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to $P$, among all such curves. Using our data structure, a query can be handled in $O(k^2 \log^3 n + k\log^4 n)$ time. More generally, a geometric tree $T$ on $n$ vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair $u$, $v$ of its vertices, a line $\ell$, and an integer $k\ge1$, one can find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to the path from $u$ to $v$ in $T$, in time $O(k^2 \mathop{polylog} n)$. For the general dimension-reduction problem, where $P$ is a curve in $\mathbb{R}^d$ ($d \ge 3$), $0 < \varepsilon_0 < 1$ is a real parameter, and a query specifies a $g$-flat $h$ ($1 \le g \le d-1$) and an integer $k \ge 1$, we construct a data structure of size $O(n\log n + f(\varepsilon_0) n)$, where $f(\varepsilon_0)=(1+1/\varepsilon_0)^{(d-1)/2}$, that allows us to find a curve $Q$ on $h$ with at most $k$ vertices, whose discrete Fréchet distance to $P$ is at most $1+\varepsilon_0$ times the distance of $Q^*$ to $P$, where $Q^*$ is such a curve that minimizes the distance to $P$. The query handling time is $O(f(\varepsilon_0) k^2 \log^2 n)$.

发表机构

• Department of Computer Science and Engineering, Tandon School of Engineering, New York University（纽约大学坦顿工程学院计算机科学系）
• The Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev（内盖夫本古里安大学斯坦计算机与信息科学学院）

机构由 AI 辅助整理，请以论文原文为准。

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21 pages, 1 figure