On the Approximate Nearest Neighbor Queries among Curves under the Fréchet Distance
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Approximate nearest neighbor search (ANNS) is a long-studied problem in computational geometry that has received considerable attentions by researchers in the community. In this paper, we revisit the problem in the presence of curves under the Fréchet distance. Given a set P of n curves of size at most m each in ℝ^d and a real δ>0, we aim to preprocess P into a data structure so that for any given query curve Q of size k, report all curves in P whose Fréchet distances to Q are at most δ. In case that k is known in the preprocessing stage we propose a fully deterministic data structure whose space is O(n(32d^1/2/ε^3)^d(k+1) ) and can answer the (1+ε)δ-ANNS queries in O(kd) query time. Considering k as part of the query slightly changes the space to O( n(64d^1/2/ε^3)^md ) with O(kd) query time within 5(1+ε) approximation factor. We also show that our data structure could give an alternative treatment of the approximate subtrajectory range counting (ASRC) problem studied by de Berg et al. <cit.>.
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