Inserting Points Uniformly at Every Instance Sachio Teramoto1, Tetsuo Asano1, Benjamin Doerr2, and Naoki Katoh3 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), 1-1 Asahidai, Nomi, Ishikawa, 923-1292 Japan, {s-teramo,t-asano}@jaist.ac.jp 2 Max Planck Institute fšur Informatik, Saarbršucken, Germany, doerr@mpi-sb.mpg.de 3 Graduate School of Engineering, Kyoto University, Kyoto-Daigaku-Katsura, Nishigyoku, Kyoto, 615-8540, Japan, naoki@archi.kyoto-u.ac.jp Abstract. A problem of arranging n points as uniformly as possible, which is equivalent to that of packing n equal and non-overlapping 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 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. 1 Introduction Circle packing problem to place n equal and non-overlapping 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,