On the existence of hyper-L triple-loop networks

Triple-loop networks with arbitrarily many minimum distance diagrams

“…But it was shown in [7,8] that these hyper-L MDD's exist only for very special parameters N , s 1 , s 2 and s 3 of the network. 1 Observe that triple-loop networks with N ≃ D 3 /27 are trivial to construct, and that, by a simple volume argument on its MDD, every triple loop network has N ≤` Figure 2.…”

“…1 Observe that triple-loop networks with N ≃ D 3 /27 are trivial to construct, and that, by a simple volume argument on its MDD, every triple loop network has N ≤` Figure 2. The "hyper-L with parameters l = 7, m = 3 and n = 2" [7]. It is a minimum distance diagram for the circulant digraph C 182 (43, 23, 25).…”

“…Rather, since the interest is in solving the routing problem, the authors consider one particular diagram, and concentrate on the algorithm to obtain it, or on studying its shape, etc. Most authors [1,2,4,6,7,10,11,13,14] always "break ties" lexicographically whenever there are two minimal paths to a vertex, so that every multi-loop network has a unique MDD for each prescribed ordering of the parameters s 1 , . .…”

“…But it was shown in [7,8] that these hyper-L MDD's exist only for very special parameters N , s 1 , s 2 and s 3 of the network.…”

“…For a small example consider the following: C 9 (1,4,7), has nine different MDD's, shown in Figure 3. Our proof of Theorem 1.2 is based in the interpretation of minimum distance diagrams in terms of initial ideals of a certain lattice ideal.…”