Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems

We present a randomized algorithm that, on input a weakly diagonally dominant symmetric n-by-n matrix A with m non-zero entries and an n-vector b, produces an x̃ such that ‖x − x̃‖ A ≤ ǫ ‖x‖ A , where Ax = b, in expected time m log n log(1/ǫ). The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya (1990). For any M0-matrix A and k ≥ 1, we construct in time m log n a preconditioner of A with at most 2(n− 1 + k) non-zero off-diagonal entries such that the preconditioned system has condition number at most (n/k) log n. If the non-zero structure of the matrix is planar, then the condition number is at most (n/k) log n log logn, and the corresponding linear system solver runs in expected time O(n log n log logn log(1/ǫ)). Similar bounds are obtained on the running time of algorithms computing approximate Fiedler vectors. This paper, and its companion [ST06], split the material that previously appeared in the paper “NearlyLinear Time Algorithms for Graph Partitioning, Graph Sparsification, and Solving Linear Systems” (Arxiv cs.DS/0310051). This paper contains the material on solving linear systems, and its companion contains the material on partitioning and sparsification. † [email protected] ‡ [email protected]