Curve Simplification and Clustering under Fréchet Distance

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 $\tilde{O}\bigl(n m^{O(kl^2)} μ^{-O(kl)} (dkl/ε)^{O((dkl/ε)\log(1/μ))}\bigr)$.