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We develop and implement in this paper a fast sparse assembly algorithm, the fundamental operation which creates a compressed matrix from raw index data. Since it is often a quite demanding and sometimes critical operation, it is of interest to design a highly efficient implementation. We show how to do this, and moreover, we show how our implementation can be parallelized to utilize the power of modern multicore computers. Our freely available code, fully Matlab compatible, achieves about a factor of 5 times in speedup on a typical 6-core machine and 10 times on a dual-socket 16 core machine compared to the built-in serial implementation

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Abstract. Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving 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 comprehensive experimental study are ambivalent. They confirm a nearlylinear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solver's running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.