Augmented ring networks

We study four augmentations of ring networks which are intended to enhance a ring's efficiency as a communication medium significantly, while increasi ng its structural complexity only modestly. Chordal rings add "shortcut" ed ges, which can be viewed as chords, to the ring. Express rings are chordal rings whose chords are routed outside the ring. Multirings append subsidiar y rings to edges of a ring and, recursively, to edges of appended subrings. Hierarchical ring networks (HRNs) append subsidiary rings to nodes of a ri ng and, recursively, to nodes of appended subrings. We show that these four modes of augmentation are very closely related: 1) Planar chordal rings, p lanar express rings, and multirings are topologically equivalent families o f networks with the "cutwidth" of an express ring translating into the "tre e depth" of its isomorphic multiring and vice versa. 2) Every depth-dHRN is a spanning subgraph of a depth-(2d - 1) multiring. 3) Every depth-d multir ing M can be embedded into a d-dimensional mesh with dilation 3 in such a w ay that some node of M resides at a corner of the mesh. 4) Every depth-d HR N H can be embedded into a d-dimensional mesh with dilation 2 in such a way that some node of H resides at a corner of the mesh. In addition to demons trating that these four augmented ring networks are grid graphs, our embedd ing results afford us close bounds on how much decrease in diameter is achi evable for a given increase in structural complexity for the networks. Spec ifically, we derive upper and lower bounds on the optimal diameters of N-no de depth-d multirings and HRNs that are asymptotically tight for large N an d d.