A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Bedini, Andrea; Jacobsen, Jesper Lykke

doi:10.1088/1751-8113/43/38/385001

Cited by 8 publications

(33 citation statements)

References 39 publications

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“…We wish to evaluate Z G (Q, v) (1.2)/(1.3) for G = G(nk, k) by a transfer matrix construction. Contrary to an often repeated but false statement, evaluating Z G (Q, v) by a transfer matrix construction is possible for any graph G, and does not require G to consist of a number of identical layers [8]. However, when G does have a layered structure-as is the case here-Z G (Q, v) can be computed by the repeated application of the same transfer matrix.…”

Section: Potts Model Transfer Matrixmentioning

confidence: 99%

A generalized Beraha conjecture for non-planar graphs

Jacobsen

Salas

2013

Nuclear Physics B

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We study the partition function Z_{G(nk,k)}(Q,v) of the Q-state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k). We study its zeros in the plane (Q,v) for 1<= k <= 7. We also consider two specializations of Z_{G(nk,k)}, namely the chromatic polynomial P_{G(nk,k)}(Q) (corresponding to v=-1), and the flow polynomial Phi_{G(nk,k)}(Q) (corresponding to v=-Q). In these two cases, we study their zeros in the complex Q-plane for 1 <= k <= 7. We pay special attention to the accumulation loci of the corresponding zeros when n -> infinity. We observe that the Berker-Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.Comment: 47 pages (LaTeX2e). Includes tex file, three sty files, and 40 Postscript figures. Minor changes from version 1. Final version published in Nucl. Phys.

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Section: Potts Model Transfer Matrixmentioning

confidence: 99%

A generalized Beraha conjecture for non-planar graphs

Jacobsen

Salas

2013

Nuclear Physics B

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“…For simplicity, we will restrict our physical results just to the computation and validation of the limiting curves using free boundary conditions in order to stay within the scope of our work, but not restrict the proposed strategy to these conditions. More general methods for computing the exact partition function of a lattice have also been proposed [32,15,33]. Bedini et.…”

Section: Related Workmentioning

confidence: 99%

“…(3) Alternatively one can compute M with a generic (q, v) method where the configuration space grows proportional to the Catalan numbers [12] or asymptotically as O(4 m ), leading to a matrix of size O(4 m × 4 m ). Indeed there are other strategies that can achieve smaller transfer matrices [13,14,15], but they assume special properties for the lattice, work only for finite graphs or need to fix the values of v and/or q in order to take any advantage. We believe it is worth studying what are the possibilities for algorithmic improvements in the generic (q, v) Catalan based approach since it is a general method applicable to any planar strip.…”

Section: Introductionmentioning

confidence: 99%

Parallel family trees for transfer matrices in the Potts model

Navarro¹,

Canfora²,

Hitschfeld³

et al. 2015

Computer Physics Communications

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The computational cost of transfer matrix methods for the Potts model is related to the question into how many ways can two layers of a lattice be connected?. Answering the question leads to the generation of a combinatorial set of lattice configurations. This set defines the configuration space of the problem, and the smaller it is, the faster the transfer matrix can be computed. The configuration space of generic (q, v) transfer matrix methods for strips is in the order of the Catalan numbers, which grows asymptotically as O(4 m ) where m is the width of the strip. Other transfer matrix methods with a smaller configuration space indeed exist but they make assumptions on the temperature, number of spin states, or restrict the structure of the lattice. In this paper we propose a parallel algorithm that uses a sub-Catalan configuration space of O(3 m ) to build the generic (q, v) transfer matrix in a compressed form. The improvement is achieved by grouping the original set of Catalan configurations into a forest of family trees, in such a way that the solution to the problem is now computed by solving the root node of each family. As a result, the algorithm becomes exponentially faster than the Catalan approach while still highly parallel. The resulting matrix is stored in a compressed form using O(3 m × 4 m ) of space, making numerical evaluation and decompression to be faster than evaluating the matrix in its O(4 m × 4 m ) uncompressed form. Experimental results for different sizes of strip lattices show that the parallel family trees (PFT) strategy indeed runs exponentially faster than the Catalan Parallel Method (CPM), especially when dealing with dense transfer matrices. In terms of parallel performance, we report strongscaling speedups of up to 5.7X when running on an 8-core shared memory machine and 28X for a 32-core cluster. The best balance of speedup and efficiency for the multi-core machine was achieved when using p = 4 processors, while for the cluster scenario it was in the range p ∈ [8, 10]. Because of the parallel capabilities of the algorithm, a large-scale execution of the parallel family trees strategy in a supercomputer could contribute to the study of wider strip lattices.

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“…al. [12] proposed a method for computing the partition function of arbitrary graphs using a treedecomposed transfer matrix. In their work, the authors obtain a sub-exponential algorithm based on arbitrary heuristics for finding a good tree decomposition of the graph.…”

Section: Preliminaries and Related Workmentioning

confidence: 99%

“…It is in this last category where most of the scientific community lies, therefore parallel implementations for multi-core machines are the ones to have the biggest impact. Latest work in the field of the Potts model has been focused on parallel probabilistic simulations [9], [10] and new sequential methods for computing the exact partition function of a lattice [11], [12], [13]. To the best of our knowledge, there has not been published research regarding parallel multi-core performance of exact transfer-matrix algorithms in the Potts model.…”

Section: Introductionmentioning

confidence: 99%

Multi-core Computation of Transfer Matrices for Strip Lattices in the Potts Model

Navarro

Hitschfeld

Canfora

2013

2013 IEEE 10th International Conference on High Performance Computing and Communications &Amp; 2013 IEEE International Conferen

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The transfer-matrix technique is a convenient way for studying strip lattices in the Potts model since the computational costs depend just on the periodic part of the lattice and not on the whole. However, even when the cost is reduced, the transfer-matrix technique is still an NP-hard problem since the time T (|V |, |E|) needed to compute the matrix grows exponentially as a function of the graph width. In this work, we present a parallel transfer-matrix implementation that scales performance under multi-core architectures. The construction of the matrix is based on several repetitions of the deletion-contraction technique, allowing parallelism suitable to multi-core machines. Our experimental results show that the multi-core implementation achieves speedups of 3.7X with p = 4 processors and 5.7X with p = 8. The efficiency of the implementation lies between 60% and 95%, achieving the best balance of speedup and efficiency at p = 4 processors for actual multi-core architectures. The algorithm also takes advantage of the lattice symmetry, making the transfer matrix computation to run up to 2X faster than its non-symmetric counterpart and use up to a quarter of the original space.

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A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Cited by 8 publications

References 39 publications

A generalized Beraha conjecture for non-planar graphs

A generalized Beraha conjecture for non-planar graphs

Parallel family trees for transfer matrices in the Potts model

Multi-core Computation of Transfer Matrices for Strip Lattices in the Potts Model

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