Routing and transmitting problems in de Bruijn networks

De Bruijn graphs, both directed and undirected, have received considerable attention as architecture for interconnection networks. In this paper, we focus on undirected de Bruijn networks of radix d and dimension 0, denoted by UB(d, 0). We first discuss the shortest-path routing problem. We present properties of the shortest paths between any two vertices of UB(d, 0) and propose two shortest-path routing algorithms, one of which has linear time complexity. Secondly, we study the transmitting problem. We establish a lower bound for the optimal transmitting time which implies in particular that the optimal transmitting problem is trivial for UB(d, 0) when d⩾5. We present a transmitting scheme on undirected binary de Bruijn networks UB(2, n) with transmitting time n-1 for n⩾5, and conjecture that the optimal transmitting time is n-1 for UB(2, n), and n for U8(3, n) and UB(4, n)