FINDING k FARTHEST PAIRS AND k CLOSEST/FARTHEST BICHROMATIC PAIRS FOR POINTS IN THE PLANE

Abstract

We study the problem of enumerating k farthest pairs for n points in the plane and the problem of enumerating k closest/farthest bichromatic pairs of n red and n blue points in the plane. We propose a new technique for geometric enumeration problems which iteratively reduces the search space by a half and provides efficient algorithms. As applications of this technique, we develop algorithms, using higher order Voronoi diagrams, for the above problems, which run in O(min{n², n log n+k^4/3log n/log^1/3 k}) time and O(n+k^4/3/log^1/3 k+k log n) space for general L_p metric with p≠2, and O(min{n², n log n+k^4/3}) time and O(n+k^4/3+k log n) space for L₂ metric. For the problem of enumerating k closest/farthest bichromatic pairs, we shall also discuss the case where we have different numbers of red and blue points. To the authors’ knowledge, no nontrivial algorithms have been known for these problems and our algorithms are faster than trivial ones when k=o(n^3/2).

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