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Open AccessArticle

Engineering a Combinatorial Laplacian Solver: Lessons Learned

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Institute of Theoretical Informatics, Karlsruhe Institute of Technology (KIT), Am Fasanengarten 5, D-76131 Karlsruhe, Germany
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Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA
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Paralution Labs UG & Co. KG, D-76571 Gaggenau, Germany
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Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in The proceedings of the 14th International Symposium on Experimental Algorithms (SEA 2015).
Academic Editor: Qianping Gu
Algorithms 2016, 9(4), 72; https://doi.org/10.3390/a9040072
Received: 9 September 2016 / Revised: 17 October 2016 / Accepted: 20 October 2016 / Published: 31 October 2016
Linear system solving is a main workhorse in applied mathematics. Recently, theoretical computer scientists contributed sophisticated algorithms for solving linear systems with symmetric diagonally-dominant (SDD) matrices in provably nearly-linear time. These algorithms are very interesting from a theoretical perspective, but their practical performance was unclear. Here, we address this gap. We provide the first implementation of the combinatorial solver by Kelner et al. (STOC 2013), which is appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix (which is SDD) corresponds to a graph; solving symmetric Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our experiments are ambivalent. While they confirm the predicted nearly-linear running time, the constant factors make the solver much slower for reasonable inputs than basic methods with higher asymptotic complexity. We were also not able to use the solver effectively as a smoother or preconditioner. Moreover, while spanning trees with lower stretch indeed reduce the solver’s running time, we experience again a discrepancy in practice: in our experiments, simple spanning tree algorithms perform better than those with a guaranteed low stretch. We expect that our results provide insights for future improvements of combinatorial linear solvers. View Full-Text
Keywords: Laplacian linear systems; graph algorithms; low-stretch spanning trees; electrical graph flows; algorithm engineering Laplacian linear systems; graph algorithms; low-stretch spanning trees; electrical graph flows; algorithm engineering
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Hoske, D.; Lukarski, D.; Meyerhenke, H.; Wegner, M. Engineering a Combinatorial Laplacian Solver: Lessons Learned. Algorithms 2016, 9, 72.

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