基于Fréchet距离的曲线简化与聚类
Curve Simplification and Clustering under Fréchet Distance
摘要：
我们提出了新的近似结果曲线简化和聚类下的Fréchet距离。设$T = \{τ_i : i \in [n] \}$是$R^d$中的多边形曲线，每个$m$顶点。令$l$为$[m]$中的任意整数。本文研究了一个广义曲线简化问题：给定$i \in [n]$的误差界$δ_i > 0$，找到一条至多$l$顶点的曲线$σ$，使得$i \in [n]$的$d_F(σ,τ_i) \le δ_i$.我们提出了一个算法，返回一个空输出或曲线$σ$的最$l$顶点，$d_F(σ,τ_i) \le δ_i + εδ_{\max}$的$i \in [n]$，其中$δ_{\max} = \max_{i \in [n]} δ_i$。如果输出为空，则在$i \in [n]$的$δ_i$与$τ_i$的Fréchet距离内，最多没有$l$顶点的曲线。运行时间为$\tilde{O}\bigl(n^{O(l)} m^{O(l^2)} (dl/ε)^{O(dl)}\bigr)$。该算法产生了第一个多项式时间双准则近似方案，以简化曲线$τ$到另一条曲线$σ$，其中$σ$的顶点可以是$R^d$中的任何地方，使得对于任何给定的$δ> 0$和任何固定的$α, ε\in (0,1)$，$d_F(σ,τ) \le (1+ε)δ$和$|σ| \le (1+α) \min\{|c| : d_F(c,τ) \le δ\}$。运行时间为$\tilde{O}\bigl(m^{O(1/α)} (d/(αε))^{O(d/α)}\bigr)$。通过结合我们的技术与文献中的一些先前的结果，我们得到了一个近似算法$(k,l)$ -中位数聚类。给定$T$，它计算$μ, ε\in (0,1)$ J曲线的集合$Σ$，每个$l$顶点，使得$\sum_{i \in [n]} \min_{σ\in Σ} d_F(σ,τ_i)$在最优值的因子$1+ε$内，对于任何给定的$μ, ε\in (0,1)$，概率至少为$1-μ$。运行时间为$\tilde{O}\bigl(n m^{O(kl^2)} μ^{-O(kl)} (dkl/ε)^{O((dkl/ε)\log(1/μ))}\bigr)$。
英文摘要：
We present new approximation results on curve simplification and clustering under Fréchet distance. Let $T = \{τ_i : i \in [n] \}$ be polygonal curves in $R^d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $δ_i > 0$ for $i \in [n]$, find a curve $σ$ of at most $l$ vertices such that $d_F(σ,τ_i) \le δ_i$ for $i \in [n]$. We present an algorithm that returns a null output or a curve $σ$ of at most $l$ vertices such that $d_F(σ,τ_i) \le δ_i + εδ_{\max}$ for $i \in [n]$, where $δ_{\max} = \max_{i \in [n]} δ_i$. If the output is null, there is no curve of at most $l$ vertices within a Fréchet distance of $δ_i$ from $τ_i$ for $i \in [n]$. The running time is $\tilde{O}\bigl(n^{O(l)} m^{O(l^2)} (dl/ε)^{O(dl)}\bigr)$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $τ$ to another curve $σ$, where the vertices of $σ$ can be anywhere in $R^d$, so that $d_F(σ,τ) \le (1+ε)δ$ and $|σ| \le (1+α) \min\{|c| : d_F(c,τ) \le δ\}$ for any given $δ> 0$ and any fixed $α, ε\in (0,1)$. The running time is $\tilde{O}\bigl(m^{O(1/α)} (d/(αε))^{O(d/α)}\bigr)$. By combining our technique with some previous results in the literature, we obtain an approximation algorithm for $(k,l)$-median clustering. Given $T$, it computes a set $Σ$ of $k$ curves, each of $l$ vertices, such that $\sum_{i \in [n]} \min_{σ\in Σ} d_F(σ,τ_i)$ is within a factor $1+ε$ of the optimum with probability at least $1-μ$ for any given $μ, ε\in (0,1)$. The running time is $\tilde{O}\bigl(n m^{O(kl^2)} μ^{-O(kl)} (dkl/ε)^{O((dkl/ε)\log(1/μ))}\bigr)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
