Paper by Erik D. Demaine

Reference:

Erik D. Demaine, John Iacono, and Stefan Langerman, “Proximate Point Searching”, Computational Geometry: Theory and Applications, volume 28, number 1, May 2004, pages 29–40. Special issue of selected papers from the 14th Canadian Conference on Computational Geometry, 2002.

Abstract:

In the 2D point searching problem, the goal is to preprocess n points P = {p₁, …, p_n} in the plane so that, for an online sequence of query points q₁, …, q_m, it can quickly determined which (if any) of the elements of P are equal to each query point q_i. This problem can be solved in O(log n) time by mapping the problem to one dimension. We present a data structure that is optimized for answering queries quickly when they are geometrically close to the previous successful query. Specifically, our data structure executes queries in time O(log d(q_i−1, q_i)), where d is some distance function between two points, and uses O(n log n) space. Our structure works with a variety of distance functions. In contrast, it is proved that, for some of the most intuitive distance functions d, it is impossible to obtain an O(log d(q_i−1, q_i)) runtime, or any bound that is o(log n).

Comments:

This paper is also available from ScienceDirect.

Length:

The paper is 15 pages.

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PointSearching_CCCG2002 (Proximate Point Searching)