Embedding Cartesian Products of Graphs into de Bruijn Graphs

Integration of concepts for the parallelization of image processing algorithms into parallel compiler technology. Abstract Given a Cartesian product G = G 1 : : : G m (m 2) of nontrivial connected graphs G i and the n{dimensional base B de Bruijn graph D = D B (n), it is investigated whether or not G is a spanning subgraph of D. Special attention is given to graphs G 1 : : : G m which are relevant for parallel computing, namely, to Cartesian products of paths (\grids") or cycles (\tori"). For 2{dimensional de Bruijn graphs D, we present a theorem stating that certain structural assumptions on the factors G 1 ; : : :; G m ensure that G 1 : : : G m is a spanning subgraph of D. As corollaries, we obtain results improving previous results of Heydemann, Opatrny, and Sotteau (1994) on embedding grids and tori into de Bruijn graphs. Speciically, we obtain that (i) any grid G = G 1 : : : G m is a spanning subgraph of D = D B (2) provided that jGj = jDj, and (ii) any torus G = G 1 : : : G m is a spanning subgraph of D = D B (2) provided that jGj = jDj and that the G i are cycles of even length 4. We show that these results have consequences for the case n > 2, too: for even n, we apply our results to obtain embeddings of grids and tori G into de Bruijn graphs D B (n) with dilation n=2, where the base B is a xed integer 2, and n is big enough to ensure jGj jD B (n)j. We also contrast our results for n = 2 with nonexistence results for n 3 and brieey describe experimental results in the area of parallel image processing.