Summary: Implementing 2d Tori Algorithms on de Bruijn
Internal Report 4/94
M. N¨olle and G. Schreiber
Technische Universit¨at Hamburg--Harburg
Technische Informatik I
Prof. Dr. Ing. H. Burkhardt
In this paper we discuss embeddings of 2d tori in de Bruijn networks.
To our knowledge the best known embedding of 2d tori in N node de Bruijn graphs of base
two has a dilation of log 2 N while the best theoretical lower bound is O(log log N) (). Heath
claims in () that it is not possible to embed 2d tori in de Bruijn graphs with dilation less than
O(log N ), but up to now there doesn't seem to exist a proof.
After extending the scope to de Bruijn graphs of any base, we prove that a 2d torus is a spanning
subgraph of the two dimensional de Bruijn graph. In conjunction with an embedding of one
de Bruijn graph into another of different base this provides us with an embedding of 2d tori
in fixed base B de Bruijn graphs with dilation log B (