Approximate nearest line search in high dimensions

We consider the Approximate Nearest Line Search (NLS) problem. Given a set L of N lines in the high dimensional Euclidean space Rd, the goal is to build a data structure that, given a query point q ∈ Rd, reports a line l ∈ L such that its distance to the query is within (1 + ε) factor of the distance of the closest line to the query point q. The problem is a natural generalization of the well-studied Approximate Nearest Neighbor problem for point sets (ANN), and is a natural first step towards understanding how to build efficient nearest-neighbor data structures for objects that are more complex than points. Our main result is a data structure that, for any fixed ε > 0, reports the approximate nearest line in time (d + log N + 1/ε)O(1) using O(N + d)O(1/ε2) space. This is the first high-dimensional data structure for this problem with poly-logarithmic query time and polynomial space. In contrast, the best previous data structure for this problem, due to Magen [16], required quasi-polynomial space. Up to polynomials, the bounds achieved by our data structure match the performance of the best algorithm for the approximate nearest neighbor problem for point sets.