 
Summary: Inserting Points Uniformly at Every Instance
Sachio Teramoto1, Tetsuo Asano1, Benjamin Doerr2, and Naoki Katoh3
1 School of Information Science, Japan Advanced Institute of Science and Technology (JAIST), 11 Asahidai, Nomi,
Ishikawa, 9231292 Japan, {steramo,tasano}@jaist.ac.jp
2 Max Planck Institute fšur Informatik, Saarbršucken, Germany, doerr@mpisb.mpg.de
3 Graduate School of Engineering, Kyoto University, KyotoDaigakuKatsura, Nishigyoku, Kyoto, 6158540, Japan,
naoki@archi.kyotou.ac.jp
Abstract. A problem of arranging n points as uniformly as possible, which is equivalent to that of packing
n equal and nonoverlapping circles in a unit square, is frequently asked. In this paper we generalize this
problem in such a way that points be inserted one by one with uniformity preserved at every instance. Our
criteria on 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 npoint sequence with the maximum
gap ratio bounded by 2 n/2 /( n/2 +1) in the 1dimensional 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.
1 Introduction
Circle packing problem to place n equal and nonoverlapping circles in a unit square is one of impor
tant geometric optimization problems with a number of applications and has been intensively inves
tigated [6, 1214]. It is well known that the circle packing problem is equivalent to that of placing n
points in a unit square in such a way that the minimum pairwise distance is maximized. This problem
seems to be computationally hard. In fact, no optimal solution is known for relatively large value of n,
