Christopher Melgaard | Search arXiv

arXiv2015

Gaussian Elimination with Randomized Complete Pivoting

Gaussian elimination with partial pivoting (GEPP) has long been among the most widely used methods for computing the LU factorization of a given matrix. However, this method is also known to fail for matrices that induce large element growth during the factorization process. In this paper, we propose a new scheme, Gaussian elimination with randomized complete pivoting (GERCP) for the efficient and reliable LU factorization of a given matrix. GERCP satisfies GECP (Gaussian elimination with complete pivoting) style element growth bounds with high probability, yet costs only marginally higher than GEPP. Our numerical experimental results strongly suggest that GERCP is as reliable as GECP and as efficient as GEPP for computing the LU factorization.

math.NA↗

arXiv2014

An Efficient Algorithm for Unweighted Spectral Graph Sparsification

Spectral graph sparsification has emerged as a powerful tool in the analysis of large-scale networks by reducing the overall number of edges, while maintaining a comparable graph Laplacian matrix. In this paper, we present an efficient algorithm for the construction of a new type of spectral sparsifier, the unweighted spectral sparsifier. Given a general undirected and unweighted graph $G = (V, E)$ and an integer $\ell < |E|$ (the number of edges in $E$), we compute an unweighted graph $H = (V, F)$ with $F \subset E$ and $|F| = \ell$ such that for every $x \in \mathbb{R}^{V}$ \[ {\displaystyle \frac{x^T L_G x}κ \leq x^T L_H x \leq x^T L_G x,} \] where $L_G$ and $L_H$ are the Laplacian matrices for $G$ and $H$, respectively, and $κ\geq 1$ is a slowly-varying function of $|V|, |E|$ and $\ell$. This work addresses the open question of the existence of unweighted graph sparsifiers for unweighted graphs. Additionally, our algorithm can efficiently compute unweighted graph sparsifiers for weighted graphs, leading to sparsified graphs that retain the weights of the original graphs.

cs.DS↗