434 research outputs found
Embeddings into the Pancake Interconnection Network
Article paru en 2002 dans Parallel Processing LettersInternational audienceOwing to its nice properties, the pancake is one of the Cayley graphs that were proposed as alternatives to the hypercube for interconnecting processors in parallel computers. In this paper, we present embeddings of rings, grids and hypercubes into the pancake with constant dilation and congestion. We also extend the results to similar efficient embeddings into the star graph
Graph Laplacians, Nodal Domains, and Hyperplane Arrangements
Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e. the connected components of the maximal induced subgraphs of G on which an eigenvector Ï does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of Ï in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures
Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics
Kinetically constrained lattice models of glasses introduced by Kob and
Andersen (KA) are analyzed. It is proved that only two behaviors are possible
on hypercubic lattices: either ergodicity at all densities or trivial
non-ergodicity, depending on the constraint parameter and the dimensionality.
But in the ergodic cases, the dynamics is shown to be intrinsically cooperative
at high densities giving rise to glassy dynamics as observed in simulations.
The cooperativity is characterized by two length scales whose behavior controls
finite-size effects: these are essential for interpreting simulations. In
contrast to hypercubic lattices, on Bethe lattices KA models undergo a
dynamical (jamming) phase transition at a critical density: this is
characterized by diverging time and length scales and a discontinuous jump in
the long-time limit of the density autocorrelation function. By analyzing
generalized Bethe lattices (with loops) that interpolate between hypercubic
lattices and standard Bethe lattices, the crossover between the dynamical
transition that exists on these lattices and its absence in the hypercubic
lattice limit is explored. Contact with earlier results are made via analysis
of the related Fredrickson-Andersen models, followed by brief discussions of
universality, of other approaches to glass transitions, and of some issues
relevant for experiments.Comment: 59 page
Optical control plane: theory and algorithms
In this thesis we propose a novel way to achieve global network information dissemination in which some wavelengths are reserved exclusively for global control information exchange. We study the routing and wavelength assignment problem for the special communication pattern of non-blocking all-to-all broadcast in WDM optical networks. We provide efficient solutions to reduce the number of wavelengths needed for non-blocking all-to-all broadcast, in the absence of wavelength converters, for network information dissemination. We adopt an approach in which we consider all nodes to be tap-and-continue capable thus studying lighttrees rather than lightpaths. To the best of our knowledge, this thesis is the first to consider âtap-and-continueâ capable nodes in the context of conflict-free all-to-all broadcast. The problem of all to-all broadcast using individual lightpaths has been proven to be an NP-complete problem [6]. We provide optimal RWA solutions for conflict-free all-to-all broadcast for some particular cases of regular topologies, namely the ring, the torus and the hypercube. We make an important contribution on hypercube decomposition into edge-disjoint structures. We also present near-optimal polynomial-time solutions for the general case of arbitrary topologies. Furthermore, we apply for the first time the âcactusâ representation of all minimum edge-cuts of graphs with arbitrary topologies to the problem of all-to-all broadcast in optical networks. Using this representation recursively we obtain near-optimal results for the number of wavelengths needed by the non-blocking all-to-all broadcast. The second part of this thesis focuses on the more practical case of multi-hop RWA for non- blocking all-to-all broadcast in the presence of Optical-Electrical-Optical conversion. We propose two simple but efficient multi-hop RWA models. In addition to reducing the number of wavelengths we also concentrate on reducing the number of optical receivers, another important optical resource. We analyze these models on the ring and the hypercube, as special cases of regular topologies. Lastly, we develop a good upper-bound on the number of wavelengths in the case of non-blocking multi-hop all-to-all broadcast on networks with arbitrary topologies and offer a heuristic algorithm to achieve it. We propose a novel network partitioning method based on âvirtual perfect matchingâ for use in the RWA heuristic algorithm
Properties and algorithms of the (n, k)-arrangement graphs
The (n, k)-arrangement interconnection topology was first introduced in 1992. The
(n, k )-arrangement graph is a class of generalized star graphs. Compared with the
well known n-star, the (n, k )-arrangement graph is more flexible in degree and diameter.
However, there are few algorithms designed for the (n, k)-arrangement graph
up to present. In this thesis, we will focus on finding graph theoretical properties
of the (n, k)- arrangement graph and developing parallel algorithms that run on this
network.
The topological properties of the arrangement graph are first studied. They include
the cyclic properties. We then study the problems of communication: broadcasting
and routing. Embedding problems are also studied later on. These are very
useful to develop efficient algorithms on this network.
We then study the (n, k )-arrangement network from the algorithmic point of view.
Specifically, we will investigate both fundamental and application algorithms such as
prefix sums computation, sorting, merging and basic geometry computation: finding
convex hull on the (n, k )-arrangement graph.
A literature review of the state-of-the-art in relation to the (n, k)-arrangement
network is also provided, as well as some open problems in this area
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