Multi-core computation of transfer matrices for strip lattices in the Potts model

TL;DR

This paper introduces a parallel multi-core implementation of the transfer-matrix method for strip lattices in the Potts model, significantly improving computational speed and efficiency through parallelism and symmetry exploitation.

Contribution

It presents a scalable multi-core transfer-matrix algorithm using deletion-contraction repetitions, optimizing performance and memory use for lattice computations.

Findings

Achieves up to 5.7X speedup with 8 processors

Efficiency ranges from 60% to 95% depending on processors

Lattice symmetry reduces computation time by up to 2X

Abstract

The transfer-matrix technique is a convenient way for studying strip lattices in the Potts model since the compu- tational 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 ex- ponentially 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.