Approximate Nearest Neighbor for Curves: Simple, Efficient, and Deterministic

In the \((1+{\varepsilon },r)\)-approximate near-neighbor problem for curves (ANNC) under some similarity measure \(\delta \), goal is to construct a data structure given set \(\mathcal {C}\) of that supports approximate queries: Given query curve Q, if there exists \(C\in \mathcal such (Q,C)\le r\), then return \(C'\in with (Q,C')\le (1+{\varepsilon })r\). There an efficient reduction from })\)-approximate nearest-neighbor ANNC, where in former answer })\cdot \delta (Q,C^*)\), \(C^*\) most similar Q. n curves, each consisting m points d dimensions, we ANNC uses \(n\cdot O(\frac{1}{{\varepsilon }})^{md}\) storage space and has O(md) time (for length m), between two their discrete Fréchet or dynamic warping distance. Our method simple implement, deterministic, results exponential improvement both compared all previous bounds. Further, also consider asymmetric version \(k \ll m\), obtain essentially same bounds as above, except replaced by k. Finally, apply our range counting achieve