A linear work, O(n1/6) time, parallel algorithm for solving planar Laplacians

We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector x̄ such that ||x − x̄||A ≤ ǫ, in O(n log(1/ǫ)) parallel time, doing O(n log(1/ǫ)) work, where c is any positive constant. One of the key ingredients of the solver, is an O(nk log k) work, O(k logn) time, parallel algorithm for decomposing any embedded planar graph into components of size O(k) that are delimited by O(n/ √ k) boundary edges. The result also applies to symmetric diagonally dominant matrices of planar structure.