2,042 research outputs found
Symmetric Interconnection Networks from Cubic Crystal Lattices
Torus networks of moderate degree have been widely used in the supercomputer
industry. Tori are superb when used for executing applications that require
near-neighbor communications. Nevertheless, they are not so good when dealing
with global communications. Hence, typical 3D implementations have evolved to
5D networks, among other reasons, to reduce network distances. Most of these
big systems are mixed-radix tori which are not the best option for minimizing
distances and efficiently using network resources. This paper is focused on
improving the topological properties of these networks.
By using integral matrices to deal with Cayley graphs over Abelian groups, we
have been able to propose and analyze a family of high-dimensional grid-based
interconnection networks. As they are built over -dimensional grids that
induce a regular tiling of the space, these topologies have been denoted
\textsl{lattice graphs}. We will focus on cubic crystal lattices for modeling
symmetric 3D networks. Other higher dimensional networks can be composed over
these graphs, as illustrated in this research. Easy network partitioning can
also take advantage of this network composition operation. Minimal routing
algorithms are also provided for these new topologies. Finally, some practical
issues such as implementability and preliminary performance evaluations have
been addressed
On the Complexity of Diameter and Related Problems in Permutation Groups
We prove that it is ??^?-complete to verify whether the diameter of a given permutation group G = ?A? is bounded by a unary encoded number k. This solves an open problem from a paper of Even and Goldreich, where the problem was shown to be NP-hard. Verifying whether the diameter is exactly k is complete for the class consisting of all intersections of a ??^?-language and a ??^?-language. A similar result is shown for the length of a given permutation ?, which is the minimal k such that ? can be written as a product of at most k generators from A. Even and Goldreich proved that it is NP-complete to verify, whether the length of a given ? is at most k (with k given in unary encoding). We show that it is DP-complete to verify whether the length is exactly k. Finally, we deduce from our result on the diameter that it is ??^?-complete to check whether a given finite automaton with transitions labelled by permutations from S_n produces all permutations from S_n
Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks
Given a pair of graphs with the same number of vertices, the inexact graph
matching problem consists in finding a correspondence between the vertices of
these graphs that minimizes the total number of induced edge disagreements. We
study this problem from a statistical framework in which one of the graphs is
an errorfully observed copy of the other. We introduce a corrupting channel
model, and show that in this model framework, the solution to the graph
matching problem is a maximum likelihood estimator. Necessary and sufficient
conditions for consistency of this MLE are presented, as well as a relaxed
notion of consistency in which a negligible fraction of the vertices need not
be matched correctly. The results are used to study matchability in several
families of random graphs, including edge independent models, random regular
graphs and small-world networks. We also use these results to introduce
measures of matching feasibility, and experimentally validate the results on
simulated and real-world networks
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