412 research outputs found
Notes on the connectivity of Cayley coset digraphs
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
Let be a Cayley graph of the permutation group generated by a
transposition tree on 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 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 ,
denotes the number of cycles in , and \dist_T is the distance
function in . 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 , there exists a tree on vertices, such that the
difference between the upper bound and the true diameter value is at least
.
Observe that evaluating this upper bound requires on the order of (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)
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
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 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.
Biswapped networks of the form have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of vertex-disjoint paths joining any two distinct vertices in , where is the connectivity of . In doing so, we obtain an upper bound of on the -diameter of , where is the diameter of and the -diameter. Suppose that we have a deterministic multipath source routing algorithm in an interconnection network that finds mutually vertex-disjoint paths in joining any distinct vertices and does this in time polynomial in , and (and independently of the number of vertices of ). Our constructions yield an analogous deterministic multipath source routing algorithm in the interconnection network that finds mutually vertex-disjoint paths joining any distinct vertices in so that these paths all have length bounded as above. Moreover, our algorithm has time complexity polynomial in , and . We also show that if is Hamiltonian then is Hamiltonian, and that if is a Cayley graph then is a Cayley graph
Optimal Networks from Error Correcting Codes
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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