Curve Simplification and Clustering under Fréchet Distance

Abstract: We present new approximation results on curve simplification and clustering under Fr\'echet distance. Let $T = {\tau_i : i \in [n] }$ be polygonal curves in $R<sup>d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $\delta_i &gt; 0$ for $i \in [n]$, find a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i$ for $i \in [n]$. We present an algorithm that returns a null output or a curve $\sigma$ of at most $l$ vertices such that $$d_F(\sigma,\tau_i) \le \delta_i + \epsilon\delta_{\max}$$ for $i \in [n]$, where $\delta_{\max} = \max_{i \in [n]} \delta_i$. If the output is null, there is no curve of at most $l$ vertices within a Fr\'echet distance of $\delta_i$ from $\tau_i$ for $i \in [n]$. The running time is $$\tilde{O}\bigl(n^{O(l)} m^{{O(l^2)}} (dl/\epsilon)<sup>{O(dl)}\bigr)$$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $\tau$ to another curve $\sigma$, where the vertices of $\sigma$ can be anywhere in $R<sup>d$, so that $d_F(\sigma,\tau) \le (1+\epsilon)\delta$ and $$|\sigma| \le (1+\alpha) \min{|c| : d_F(c,\tau) \le \delta}$$ for any given $\delta &gt; 0$ and any fixed $\alpha, \epsilon \in (0,1)$. The running time is $$\tilde{O}\bigl(m^{{O(1/\alpha)}} (d/(\alpha\epsilon))<sup>{O(d/\alpha)}\bigr)$$. 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 $\Sigma$ of $k$ curves, each of $l$ vertices, such that $$\sum_{i \in [n]} \min_{\sigma \in \Sigma} d_F(\sigma,\tau_i)$$ is within a factor $1+\epsilon$ of the optimum with probability at least $1-\mu$ for any given $\mu, \epsilon \in (0,1)$. The running time is $$\tilde{O}\bigl(n m^{{O(kl^2)}} \mu^{-O(kl)} (dkl/\epsilon)<sup>{O((dkl/\epsilon)\log(1/\mu))}\bigr)$$.