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# Approximate Nearest Neighbor for Curves – Simple, Efficient, and Deterministic

### Abstract:

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

Based on joint work with Arnold Filtser and Omrit Filtser.