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/ɛ 2) edges that provides a strong approximation of G. Namely, for all vectors x and any ɛ> 0, we have (1 − ɛ)x T LGx ≤ x T LHx ≤ (1 + ɛ)x T LGx, where LG and LH are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in Õ(m log2 n) time, an O(log n) factor faster than before. The second algorithm runs in Õ(m log n) time and generates a sparsifier with Õ(n log3 n) edges. The third algorithm applies to graphs where m> n log 5 n and runs in Õ(m logm/n log5 n n) time. In the range where m> n1+r for some constant r this becomes Õ(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices
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