Clustering methods for geometric objects and applications to design problems

Clustering of geometric objects is a very familiar and important problem in many different areas of applications as well as in the theoretical foundation of some modern fields of computer science. This paper describes how design problems, especially the design of an assembly line, can be transformed into a clustering problem. In order to solve the problem for large sizes of input data we introduce a structure, called Voronoi Tree, which applied to our real world data (assembly line design) did not only reduce the time to get a feasible design of an assembly line dramatically, but additionally increased the value of the design by more than 30% (in comparison with standard design methods). In addition to this we introduce a clustering method which is of interest for those applications which can be transformed to planar clustering problems. In this particular case it is possible to compute an (hierarchically) optimized clustering with resp. to a large class of clustering measures in timeO(nn^1/2log³ n+U _F(n)nn^1/2+P _F(n)) n: number of points;U _F(n), P_F(n) dependent on the chosen clustering measure].