30 research outputs found
Recursive circulants and their embeddings among hypercubes
AbstractWe propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N,d) is defined to be a circulant graph with N nodes and jumps of powers of d. G(N,d) is node symmetric, and has some strong hamiltonian properties. G(N,d) has a recursive structure when N=cdm, 1⩽c<d. We develop a shortest-path routing algorithm in G(cdm,d), and analyze various network metrics of G(cdm,d) such as connectivity, diameter, mean internode distance, and visit ratio. G(2m,4), whose degree is m, compares favorably to the hypercube Qm. G(2m,4) has the maximum possible connectivity, and its diameter is ⌈(3m−1)/4⌉. Recursive circulants have interesting relationship with hypercubes in terms of embedding. We present expansion one embeddings among recursive circulants and hypercubes, and analyze the costs associated with each embedding. The earlier version of this paper appeared in Park and Chwa (Proc. Internat. Symp. Parallel Architectures, Algorithms and Networks ISPAN’94, Kanazawa, Japan, December 1994, pp. 73–80)
Long cycles and paths in distance graphs
AbstractFor n∈N and D⊆N, the distance graph PnD has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, PnD has a component of order at least n−cD if and only if for every n≥cD+3, PnD has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result
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Hamiltonian Decompositions of Regular Topology Networks with Convergence Routing
This paper introduces new methods to construct multiple virtual rings for loss-free routing of non-reserved bursty data in high-speed environments such as ATM LANs. The routing algorithm on multiple virtual rings is convergence routing which combines the actual routing decision with the internal flow control state. Multiple virtual rings are obtained on the hypercube and the circulant networks such that each virtual ring is hamiltonian, and are mutually edge-disjoint. It is shown that multiple virtual rings improve (i) the bound on the length of routing, and (ii) the fault tolerance. On the circulant graphs, necessary and sufficient conditions for hamiltonian decomposition is established. On the hypercube, three algorithms are designed for an N-node hypercube with even dimension: (i) an O(N) time algorithm to find two edge-disjoint hamiltonian circuits, (ii) an O(N log N) time algorithm to find I01N hamiltonian circuits with only E ~ 0.1 common edges, and (iii) a recursive algorithm for the hamiltonian decomposition of the hypercube with dimension power of two. It is shown analytically, and verified by simulations on the circulants that with the d virtual ring embeddings, a bound of O( N / d) is established on the maximum length of routing
Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning
The paper introduces the application of information geometry to describe the
ground states of Ising models by utilizing parity-check matrices of cyclic and
quasi-cyclic codes on toric and spherical topologies. The approach establishes
a connection between machine learning and error-correcting coding. This
proposed approach has implications for the development of new embedding methods
based on trapping sets. Statistical physics and number geometry applied for
optimize error-correcting codes, leading to these embedding and sparse
factorization methods. The paper establishes a direct connection between DNN
architecture and error-correcting coding by demonstrating how state-of-the-art
architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range
arena can be equivalent to of block and convolutional LDPC codes (Cage-graph,
Repeat Accumulate). QC codes correspond to certain types of chemical elements,
with the carbon element being represented by the mixed automorphism
Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and
the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix
are elaborated upon in detail. The Quantum Approximate Optimization Algorithm
(QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous
to the back-propagation loss function landscape in training DNNs. This
similarity creates a comparable problem with TS pseudo-codeword, resembling the
belief propagation method. Additionally, the layer depth in QAOA correlates to
the number of decoding belief propagation iterations in the Wiberg decoding
tree. Overall, this work has the potential to advance multiple fields, from
Information Theory, DNN architecture design (sparse and structured prior graph
topology), efficient hardware design for Quantum and Classical DPU/TPU (graph,
quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text
overlap with arXiv:2109.08184 by other author
Maximal diameter of integral circulant graphs
Integral circulant graphs are proposed as models for quantum spin networks
that permit a quantum phenomenon called perfect state transfer. Specifically,
it is important to know how far information can potentially be transferred
between nodes of the quantum networks modelled by integral circulant graphs and
this task is related to calculating the maximal diameter of a graph. The
integral circulant graph has the vertex set and vertices and are adjacent if ,
where . Motivated by the result on
the upper bound of the diameter of given in [N. Saxena, S. Severini,
I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic
quantum dynamics}, International Journal of Quantum Information 5 (2007),
417--430], according to which represents one such bound, in this paper
we prove that the maximal value of the diameter of the integral circulant graph
of a given order with its prime factorization
, is equal to or , where
, depending on whether
or not, respectively. Furthermore, we show that, for a given
order , a divisor set with can always be found such that
this bound is attained. Finally, we calculate the maximal diameter in the class
of integral circulant graphs of a given order and cardinality of the
divisor set and characterize all extremal graphs. We actually show
that the maximal diameter can have the values , , and
depending on the values of and . This way we further improve the upper
bound of Saxena, Severini and Shparlinski and we also characterize all graphs
whose diameters are equal to , thus generalizing a result in that
paper.Comment: 29 pages, 1 figur
Hyperdeterminantal computation for the Laughlin wave function
International audienceThe decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing to compute one coefficient of the development without computing the others. Thanks to a program in {\tt C}, we performed the calculation for the square of the Vandermonde up to an alphabet of eleven lettres
The directed Oberwolfach problem with variable cycle lengths: a recursive construction
The directed Oberwolfach problem OP asks whether the
complete symmetric digraph , assuming , admits a
decomposition into spanning subdigraphs, each a disjoint union of directed
cycles of lengths . We hereby describe a method for
constructing a solution to OP given a solution to
OP, for some , if certain conditions on
are satisfied. This approach enables us to extend a solution
for OP into a solution for
OP, as well as into a solution for
OP, where denotes copies of 2, provided is sufficiently large.
In particular, our recursive construction allows us to effectively address
the two-table directed Oberwolfach problem. We show that OP has
a solution for all , with a definite exception of
and a possible exception in the case that , is even,
and . It has been shown previously that OP has
a solution if is odd, and that OP has a solution if and
only if .
In addition to solving many other cases of OP, we show that when , OP has a solution if and
only if