On the Mathematics of Data Centre Network Topologies

The theory of combinatorial designs has recently been used in order to build switch-centric data centre networks incorporating a large number of servers, in comparison with the popular Fat-Tree data centre network. The construction employed, called the 3-step method, revolves around an appropriately chosen (but relatively small) bipartite graph and a transversal design. In this paper, we clarify and extend these recent results. In particular, we prove the following path diversity results: in a one-to-one context, we prove that in these data centre networks there are pairwise link-disjoint paths joining all the servers adjacent to some switch with all the servers adjacent to any other switch so that we retain control of the path lengths (these results are optimal in terms of the numbers of paths constructed and we prove that we have a wide choice of bipartite graph and transversal design to which we can apply the 3-step method); and in a oneto-many context, we prove that there are pairwise link-disjoint paths from all the servers adjacent to some switch to any identically-sized collection of target servers where these target servers need not be adjacent to the same switch (again, we keep control of the path lengths). Our constructions and analysis are undertaken on bipartite graphs with the applications to data centre networks being easily derived. Our results strengthen the overall competitiveness of data centre networks constructed using the 3-step method, in comparison with FatTree data centre networks, and, more generally, show the potential of results and methodologies from combinatorics to data centre network design.