Near-optimal broadcast in all-port wormhole-routed hypercubes using error-correcting codes

A new broadcasting method is presented for hypercubes with wormhole routing mechanism. The communication model assumed allows an n-dimensional hypercube to have at most n concurrent I/O communication along its ports. It assumes a distance insensitivity of (n + 1) with no intermediate reception capability for the nodes. The approach is based on determination of the set of nodes called stations in the hypercube. Once stations are identified, node disjoint paths are formed from the source to all stations. The broadcasting is accomplished first by sending the message to all stations, which will inform the rest of the nodes. To establish node-disjoint paths between the source node and all stations, we introduce a new routing strategy. We prove that multicasting can be done in one routing step as long as the number of destination nodes are at most n in an n-dimensional hypercube. The number of broadcasting steps using our routing is equal to or smaller than that obtained in an earlier work; this number is optimal for all hypercube dimensions n ≤ 12, except for n = 10

From Hall's Matching Theorem to Optimal Routing on Hypercubes

AbstractWe introduce a concept of so-called disjoint ordering for any collection of finite sets. It can be viewed as a generalization of a system of distinctive representatives for the sets. It is shown that disjoint ordering is useful for network routing. More precisely, we show that Hall's “marriage” condition for a collection of finite sets guarantees the existence of a disjoint ordering for the sets. We next use this result to solve a problem in optimal routing on hypercubes. We give a necessary and sufficient condition under which there are internally node-disjoint paths each shortest from a source node to any others(s⩽n) target nodes on ann-dimensional hypercube. When this condition is not necessarily met, we show that there are always internally node-disjoint paths each being either shortest or near shortest, and the total length is minimum. An efficient algorithm is also given for constructing disjoint orderings and thus disjoint short paths. As a consequence, Rabin's information disposal algorithm may be improved

Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes

The $n$-dimensional hypercube network $Q_n$ is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional locally twisted cube, denoted by $LTQ_n$, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as $Q_n$. One advantage of $LTQ_n$ is that the diameter is only about half of the diameter of $Q_n$. Recently, some interesting properties of $LTQ_n$ were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in the locally twisted cube $LTQ_n$, for any integer $n\geqslant 4$. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the locally twisted cube.Comment: 7 pages, 4 figure