Inserting Points Uniformly at Every Instance

Authors:
Sachio Teramoto;Tetsuo Asano;Naoki Katoh;Benjamin Doerr
Affiliations:
The authors are with the School of Information Science, JAIST, Nomi-shi, 923--1292 Japan. E-mail: s-teramo@jaist.ac.jp,;The authors are with the School of Information Science, JAIST, Nomi-shi, 923--1292 Japan. E-mail: s-teramo@jaist.ac.jp,;The author is with the Graduate School of Kyoto University, Kyoto-shi, 615--8540 Japan.,;The author is with Max-Planck-Institut für Informatik, Saarbrücken, Germany.
Venue:
IEICE - Transactions on Information and Systems
Year:
2006

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Abstract

Arranging n points as uniformly as possible is a frequently occurring problem. It is equivalent to packing n equal and non-overlapping circles in a unit square. In this paper we generalize this problem in such a way that points are inserted one by one with uniformity preserved at every instance. Our criterion for uniformity is to minimize the gap ratio (which is the maximum gap over the minimum gap) at every point insertion. We present a linear time algorithm for finding an optimal n-point sequence with the maximum gap ratio bounded by 2⌊n/2⌋/(⌊n/2⌋+1) in the 1-dimensional case. We describe how hard the same problem is for a point set in the plane and propose a local search heuristics for finding a good solution.