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)2ⁿ/(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.