Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices

Ioannis Koutis , Alex Levin (1) , Richard Peng

1 Department of mathematics

We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/epsilon^2) edges that provides a strong approximation of G. Namely, for all vectors x and any epsilon>0, we have (1-epsilon) x^T L_G x <= x^T L_H x <= (1+epsilon) x^T L_G x, where L_G and L_H are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in tilde{O}(m log^2 n) time, an O(log n) factor faster than before. The second algorithm runs in tilde{O}(m log n) time and generates a sparsifier with tilde{O}(n log^3 n) edges. The third algorithm applies to graphs where m>n log^5 n and runs in tilde{O}(m log_{m/ n log^5 n} n time. In the range where m>n^{1+r} for some constant r this becomes softO(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.

Mots clés: Spectral sparsification; linear system solving

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Ioannis Koutis, Alex Levin, Richard Peng. Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.266-277. ⟨hal-00678205⟩