Curve Simplification and Clustering under Fréchet Distance
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We present new approximation results on curve simplification and clustering under Fréchet distance. Let T = {τ_i : i ∈ [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 ∈ [n], find a curve σ of at most l vertices such that d_F(σ,τ_i) ≤δ_i for i ∈ [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) ≤δ_i + ϵδ_max for i ∈ [n], where δ_max = max_i ∈ [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 ∈ [n]. The running time is Õ(n^O(l) m^O(l^2) (dl/ϵ)^O(dl)). 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) ≤ (1+ϵ)δ and |σ| ≤ (1+α) min{|c| : d_F(c,τ) ≤δ} for any given δ > 0 and any fixed α, ϵ∈ (0,1). The running time is Õ(m^O(1/α) (d/(αϵ))^O(d/α)). 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 ∑_i ∈ [n]min_σ∈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∈ (0,1). The running time is Õ(n m^O(kl^2)μ^-O(kl) (dkl/ϵ)^O((dkl/ϵ)log(1/μ))).
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