Approximate Nearest Neighbor for Polygonal Curves under Fréchet Distance

Abstract: We propose $\kappa$-approximate nearest neighbor (ANN) data structures for $n$ polygonal curves under the Fr\'{e}chet distance in $\mathbb{R}<sup>d$, where $\kappa \in {1+\varepsilon,3+\varepsilon}$ and $d \geq 2$. We assume that every input curve has at most $m$ vertices, every query curve has at most $k$ vertices, $k \ll m$, and $k$ is given for preprocessing. The query times are $$\tilde{O}(k(mn)^{{0.5+\varepsilon}/\varepsilon^d+} k(d/\varepsilon)<sup>{O(dk)})$$ for $(1+\varepsilon)$-ANN and $$\tilde{O}(k(mn)<sup>{0.5+\varepsilon}/\varepsilon<sup>d)$$ for $(3+\varepsilon)$-ANN. The space and expected preprocessing time are $$\tilde{O}(k(mnd<sup>d/\varepsilon<sup>d)<sup>{O(k+1/\varepsilon<sup>2)})$$ in both cases. In two and three dimensions, we improve the query times to $$O(1/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$$ for $(1+\varepsilon)$-ANN and $\tilde{O}(k)$ for $(3+\varepsilon)$-ANN. The space and expected preprocessing time improve to $$O(mn/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$$ in both cases. For ease of presentation, we treat factors in our bounds that depend purely on $d$ as~$O(1)$. The hidden polylog factors in the big-$\tilde{O}$ notation have powers dependent on $d$.