Deadlock Free Message Routing in Multiprocessor Interconnection Networks

A deadlock-free routing algorithm can be generated for arbitrary interconnection networks using the concept of virtual channels. A necessary and sufficient condition for deadlockfree routing is the absence of cycles in the channel dependency graph. Given an arbitrary network and a routing function, the cycles of the channel dependency graph can be removed by splitting physical channels into groups of virtual channels. This method is used to develop deadlock-free routing algorithms for k-ary n-cubes, for cube connected cycles, and for shuffleexchange networks

Recursive cubes of rings as models for interconnection networks

We study recursive cubes of rings as models for interconnection networks. We first redefine each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools. We give an algorithm for computing shortest paths and the distance between any two vertices in recursive cubes of rings, and obtain the exact value of their diameters. We obtain sharp bounds on the Wiener index, vertex-forwarding index, edge-forwarding index and bisection width of recursive cubes of rings. The cube-connected cycles and cube-of-rings are special recursive cubes of rings, and hence all results obtained in the paper apply to these well-known networks

Deadlock-Free Message Routing in Multiprocessor Interconnection Networks

A deadlock-free routing algorithm can be generated for arbitrary interconnection networks using the concept of virtual channels. A necessary and sufficient condition for deadlockfree routing is the absence of cycles in the channel dependency graph. Given an arbitrary network and a routing function, the cycles of the channel dependency graph can be removed by splitting physical channels into groups of virtual channels. This method is used to develop deadlock-free routing algorithms for k-ary n-cubes, for cube connected cycles, and for shuffle? exchange networks. (This is a revised version of 5206-tr-86

Cycle structure of percolation on high-dimensional tori

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdos-Renyi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus (Z/rZ)^d. While percolation clusters naturally have many short cycles, we show that the long cycles, i.e., cycles that pass through the boundary of the cube of width r/4 centered around each of their vertices, have length of order r^{d/3}, as on the critical Erdos-Renyi random graph. On the Erdos-Renyi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by Addario-Berry, Broutin and Goldschmidt (arXiv:0908.3629). Our proofs crucially rely on various new estimates of probabilities of the existence of open paths in critical Bernoulli percolation on Z^d with constraints on their lengths. We believe these estimates are interesting in their own right.Comment: To appear in AIHP; Major changes in Sections 1-4: new definition of long cycles; Theorem 1.2 is stronger than before, its proof is shortened; Proposition 2.1 is changed, since earlier one was not correct; proofs of Propositions 3.1 and 3.2 and Theorem 1.4(a) are modified; new Theorem 1.6 is included; auxiliary Theorem 1.5 of earlier version is not needed anymore, so it is delete

Cycles and components in geometric graphs: adjacency operator approach

Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, $d$-dimensional unit cube$[0; 1]$^d. Cycles are enumerated, sizes of maximal connected compo- nents are computed, and closed formulas are obtained for graph circumfer- ence and girth. Expected numbers of $k$-cycles, expected sizes of maximal components, and expected circumference and girth are also computed by considering powers of adjacency operators

Embedding cube-connected cycles graphs into faulty hypercubes

We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.