Steiner minimal trees: An introduction, parallel computation, and future work

Abstract

Given a set of N cities, construct a connected network which has minimum length. The problem is simple enough, but the catch is that you are allowed to add junctions in your network. Therefore, the problem becomes how many extra junctions should be added and where should they be placed so as to minimize the overall network length. This intriguing optimization problem is known as the Steiner minimal tree (SMT) problem, where the junctions that are added to the network are called Steiner points. This chapter presents a brief overview of the problem, presents an approximation algorithm which performs very well, then reviews the computational algorithms implemented for this problem. The foundation of this chapter is a parallel algorithm for the generation of what Pawel Winter termed T-list and its implementation. This generation of T-list is followed by the extraction of the proper answer. When Winter developed his algorithm, the time for extraction dominated the overall computation time. After Cockayne and Hewgill's work, the time to generate Tlist dominated the overall computation time. The parallel algorithms presented here were implemented in a program called PARSTEINER94, and the results show that the time to generate T-list has now been cut by an order of magnitude. So now the extraction time once again dominates the overall computation time. This chapter then concludes with the characterization of SMTs for certain size grids. Beginning with the known characterization of the SMT for a 2 × m grid, a grammar with rewrite rules is presented for characterizations of SMTs for 3 × m, 4 × m, 5 × m, 6 × m, and 7 × m grids.