25 research outputs found
Interconnection Networks Embeddings and Efficient Parallel Computations.
To obtain a greater performance, many processors are allowed to cooperate to solve a single problem. These processors communicate via an interconnection network or a bus. The most essential function of the underlying interconnection network is the efficient interchanging of messages between processes in different processors. Parallel machines based on the hypercube topology have gained a great respect in parallel computation because of its many attractive properties. Many versions of the hypercube have been introduced by many researchers mainly to enhance communications. The twisted hypercube is one of the most attractive versions of the hypercube. It preserves the important features of the hypercube and reduces its diameter by a factor of two. This dissertation investigates relations and transformations between various interconnection networks and the twisted hypercube and explore its efficiency in parallel computation. The capability of the twisted hypercube to simulate complete binary trees, complete quad trees, and rings is demonstrated and compared with the hypercube. Finally, the fault-tolerance of the twisted hypercube is investigated. We present optimal algorithms to simulate rings in a faulty twisted hypercube environment and compare that with the hypercube
Automorphisms generating disjoint Hamilton cycles in star graphs
In the first part of the thesis we define an automorphism φn for each star graph
Stn of degree n − 1, which yields permutations of labels for the edges of Stn
taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations
into permutation cycles, we are able to identify edge-disjoint Hamilton cycles
that are automorphic images of a known two-labelled Hamilton cycle H1 2(n)
in Stn. Our main result is an improvement from the existing lower bound of
bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number
of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the
improvement is from bn/8c to bn/5c. We extend this result to the cases when n
is the power of a prime other than 3 and 7.
The second part of the thesis studies ‘symmetric’ collections of edge-disjoint
Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under
general label-mapping automorphisms. We show that, for all even n, there exists
a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot
have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus,
Stn is not symmetrically Hamilton decomposable if n is not prime. We also give
cases of even n, in terms of Carmichael’s reduced totient function λ, for which
‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated
from H1 2(n) by a single automorphism, can and cannot attain the optimum
bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a
power of 2, then Stn has a spanning subgraph with more than half of the edges
of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains
an open problem as to whether the bϕ(n)/2c can be achieved for symmetric
collections, but we are able to show that, for certain odd n, a Ï•(n)/4 bound is
achievable and optimal for strongly symmetric collections.
The search for edge-disjoint Hamilton cycles in star graphs is important for the
design of interconnection network topologies in computer science. All our results
improve on the known bounds for numbers of any kind of edge-disjoint Hamilton
cycles in star graphs
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Graph Coverings with Few Eigenvalues or No Short Cycles
This thesis addresses the extent of the covering graph construction. How much must a cover X resemble the graph Y that it covers? How much can X deviate from Y? The main statistics of X and Y which we will measure are their regularity, the spectra of their adjacency matrices, and the length of their shortest cycles. These statistics are highly interdependent and the main contribution of this thesis is to advance our understanding of this interdependence. We will see theorems that characterize the regularity of certain covering graphs in terms of the number of distinct eigenvalues of their adjacency matrices. We will see old examples of covers whose lack of short cycles is equivalent to the concentration of their spectra on few points, and new examples that indicate certain limits to this equivalence in a more general setting. We will see connections to many combinatorial objects such as regular maps, symmetric and divisible designs, equiangular lines, distance-regular graphs, perfect codes, and more. Our main tools will come from algebraic graph theory and representation theory. Additional motivation will come from topological graph theory, finite geometry, and algebraic topology
Small-world interconnection networks for large parallel computer systems
The use of small-world graphs as interconnection networks of multicomputers is proposed and analysed in this work. Small-world interconnection networks are constructed by adding (or modifying) edges to an underlying local graph. Graphs with a rich local structure but with a large diameter are shown to be the most suitable candidates for the underlying graph. Generation models based on random and deterministic wiring processes are proposed and analysed. For the random case basic properties such as degree, diameter, average length and bisection width are analysed, and the results show that a fast transition from a large diameter to a small diameter is experienced when the number of new edges introduced is increased. Random traffic analysis on these networks is undertaken, and it is shown that although the average latency experiences a similar reduction, networks with a small number of shortcuts have a tendency to saturate as most of the traffic flows through a small number of links. An analysis of the congestion of the networks corroborates this result and provides away of estimating the minimum number of shortcuts required to avoid saturation. To overcome these problems deterministic wiring is proposed and analysed. A Linear Feedback Shift Register is used to introduce shortcuts in the LFSR graphs. A simple routing algorithm has been constructed for the LFSR and extended with a greedy local optimisation technique. It has been shown that a small search depth gives good results and is less costly to implement than a full shortest path algorithm. The Hilbert graph on the other hand provides some additional characteristics, such as support for incremental expansion, efficient layout in two dimensional space (using two layers), and a small fixed degree of four. Small-world hypergraphs have also been studied. In particular incomplete hypermeshes have been introduced and analysed and it has been shown that they outperform the complete traditional implementations under a constant pinout argument. Since it has been shown that complete hypermeshes outperform the mesh, the torus, low dimensional m-ary d-cubes (with and without bypass channels), and multi-stage interconnection networks (when realistic decision times are accounted for and with a constant pinout), it follows that incomplete hypermeshes outperform them as well
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum