412 research outputs found

    Notes on the connectivity of Cayley coset digraphs

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    Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

    Diameter of Cayley graphs of permutation groups generated by transposition trees

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    Let Γ\Gamma be a Cayley graph of the permutation group generated by a transposition tree TT on nn vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph Γ\Gamma is bounded as \diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}, where the maximization is over all permutations π\pi, c(π)c(\pi) denotes the number of cycles in π\pi, and \dist_T is the distance function in TT. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every nn, there exists a tree on nn vertices, such that the difference between the upper bound and the true diameter value is at least n4n-4. Observe that evaluating this upper bound requires on the order of n!n! (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) nn computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on nn vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value β\beta computed by our algorithm happens to also be an upper bound on the diameter, i.e. \diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535

    A multipath analysis of biswapped networks.

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    Biswapped networks of the form Bsw(G)Bsw(G) have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of κ+1\kappa+1 vertex-disjoint paths joining any two distinct vertices in Bsw(G)Bsw(G), where κ1\kappa\geq 1 is the connectivity of GG. In doing so, we obtain an upper bound of max{2Δ(G)+5,Δκ(G)+Δ(G)+2}\max\{2\Delta(G)+5,\Delta_\kappa(G)+\Delta(G)+2\} on the (κ+1)(\kappa+1)-diameter of Bsw(G)Bsw(G), where Δ(G)\Delta(G) is the diameter of GG and Δκ(G)\Delta_\kappa(G) the κ\kappa-diameter. Suppose that we have a deterministic multipath source routing algorithm in an interconnection network GG that finds κ\kappa mutually vertex-disjoint paths in GG joining any 22 distinct vertices and does this in time polynomial in Δκ(G)\Delta_\kappa(G), Δ(G)\Delta(G) and κ\kappa (and independently of the number of vertices of GG). Our constructions yield an analogous deterministic multipath source routing algorithm in the interconnection network Bsw(G)Bsw(G) that finds κ+1\kappa+1 mutually vertex-disjoint paths joining any 22 distinct vertices in Bsw(G)Bsw(G) so that these paths all have length bounded as above. Moreover, our algorithm has time complexity polynomial in Δκ(G)\Delta_\kappa(G), Δ(G)\Delta(G) and κ\kappa. We also show that if GG is Hamiltonian then Bsw(G)Bsw(G) is Hamiltonian, and that if GG is a Cayley graph then Bsw(G)Bsw(G) is a Cayley graph

    Optimal Networks from Error Correcting Codes

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    To address growth challenges facing large Data Centers and supercomputing clusters a new construction is presented for scalable, high throughput, low latency networks. The resulting networks require 1.5-5 times fewer switches, 2-6 times fewer cables, have 1.2-2 times lower latency and correspondingly lower congestion and packet losses than the best present or proposed networks providing the same number of ports at the same total bisection. These advantage ratios increase with network size. The key new ingredient is the exact equivalence discovered between the problem of maximizing network bisection for large classes of practically interesting Cayley graphs and the problem of maximizing codeword distance for linear error correcting codes. Resulting translation recipe converts existent optimal error correcting codes into optimal throughput networks.Comment: 14 pages, accepted at ANCS 2013 conferenc
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