*Forma,* Vol. 4 (No. 2), pp. 157-167, 2001

*Original Paper*

## Constellation Problem in 1D Universe—Size Distribution of the Clusters by Nearest Neighbor Linking

Munekazu Sakamoto and Mikio Takagi

Center for Function- Oriented Electronics, Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan

(Received July 28, 1989; Accepted August 30, 1989)

**Keywords: **
Nearest Neighbor, Clustering, Distribution Function, Constellation, Combinatorics

**Abstract. **
If a set of points (stars) on a line (universe) is given, and every point is connected to its nearest point, then there emerges a group of clusters (1 D constellations). If the stars are randomly scattered in a ring universe, the probability of an *n*-star constellation is 3(*n*-l)(*n*+2)2^{n}/(*n*+3)! for *n* 2. The average number of stars in a constellation in an *m*-dimensional universe is also shown to be from 3 to 4 as the dimension increases.