High performance lattice reduction on heterogeneous computing platform

The final publication is available at Springer via http://dx.doi.org/10.1007/s11227-014-1201-2The lattice reduction (LR) technique has become very important in many engineering fields. However, its high complexity makes difficult its use in real-time applications, especially in applications that deal with large matrices. As a solution, the modified block LLL (MB-LLL) algorithm was introduced, where several levels of parallelism were exploited: (a) fine-grained parallelism was achieved through the cost-reduced all-swap LLL (CR-AS-LLL) algorithm introduced together with the MB-LLL by Jzsa et al. (Proceedings of the tenth international symposium on wireless communication systems, 2013) and (b) coarse-grained parallelism was achieved by applying the block-reduction concept presented by Wetzel (Algorithmic number theory. Springer, New York, pp 323-337, 1998). In this paper, we present the cost-reduced MB-LLL (CR-MB-LLL) algorithm, which allows to significantly reduce the computational complexity of the MB-LLL by allowing the relaxation of the first LLL condition while executing the LR of submatrices, resulting in the delay of the Gram-Schmidt coefficients update and by using less costly procedures during the boundary checks. The effects of complexity reduction and implementation details are analyzed and discussed for several architectures. A mapping of the CR-MB-LLL on a heterogeneous platform is proposed and it is compared with implementations running on a dynamic parallelism enabled GPU and a multi-core CPU. The mapping on the architecture proposed allows a dynamic scheduling of kernels where the overhead introduced is hidden by the use of several CUDA streams. Results show that the execution time of the CR-MB-LLL algorithm on the heterogeneous platform outperforms the multi-core CPU and it is more efficient than the CR-AS-LLL algorithm in case of large matrices.Financial support for this study was provided by grants TAMOP-4.2.1./B-11/2/KMR-2011-0002, TAMOP-4.2.2/B-10/1-2010-0014 from the Pazmany Peter Catholic University, European Union ERDF, Spanish Government through TEC2012-38142-C04-01 project and Generalitat Valenciana through PROMETEO/2009/013 project.Jozsa, CM.; Domene Oltra, F.; Vidal Maciá, AM.; Piñero Sipán, MG.; González Salvador, A. (2014). High performance lattice reduction on heterogeneous computing platform. Journal of Supercomputing. 70(2):772-785. https://doi.org/10.1007/s11227-014-1201-2S772785702Józsa CM, Domene F, Piñero G, González A, Vidal AM (2013) Efficient GPU implementation of lattice-reduction-aided multiuser precoding. 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Commutative Algorithms Approximate the LLL-distribution

Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local Lemma (LLL), a series of works have exploited a key ingredient of the original analysis, the witness tree lemma, in order to: derive deterministic, parallel and distributed algorithms for the LLL, to estimate the entropy of the output distribution, to partially avoid bad events, to deal with super-polynomially many bad events, and even to devise new algorithmic frameworks. Meanwhile, a parallel line of work, has established tools for analyzing stochastic local search algorithms motivated by the LLL that do not fall within the Moser-Tardos framework. Unfortunately, the aforementioned results do not transfer to these more general settings. Mainly, this is because the witness tree lemma, provably, no longer holds. Here we prove that for commutative algorithms, a class recently introduced by Kolmogorov and which captures the vast majority of LLL applications, the witness tree lemma does hold. Armed with this fact, we extend the main result of Haeupler, Saha, and Srinivasan to commutative algorithms, establishing that the output of such algorithms well-approximates the LLL-distribution, i.e., the distribution obtained by conditioning on all bad events being avoided, and give several new applications. For example, we show that the recent algorithm of Molloy for list coloring number of sparse, triangle-free graphs can output exponential many list colorings of the input graph