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We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental sparsifier G with n-1 + m/k edges, such that the condition number of G with G is bounded above by O (k² n), with probability 1-p. The algorithm runs in time O ( (m n + n²n) (1/p) ). As a result, we obtain an algorithm that on input of an n n symmetric diagonally dominant matrix A with m non-zero entries and a vector b, computes a vector x satisfying ||x-A^+b||A
Koutis et al. (Mon,) studied this question.