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 <i>Ax</i> = <i>b</i>, where <i>A</i> is the Laplacian of any planar graph with <i>n</i> nodes, the algorithm produces a vector <i>x</i> such that ||<i>x</i>--<i>x</i>||<i>A</i> ≤ ε, in <i>O</i>(<i>n</i><sup>1/6+</sup><i>c</i>log(1/ε)) parallel time, doing <i>O</i>(<i>n</i>log(1/ε)) work, where <i>c</i> is any positive constant. One of the key ingredients of the solver, is an <i>O</i>(<i>nk</i>log<sup>2</sup><i>k</i>) work, <i>O</i>(<i>k</i>log<i>n</i>) time, parallel algorithm for decomposing any embedded planar graph into components of size <i>O</i>(<i>k</i>) that are delimited by <i>O</i>(<i>n</i>/√<i>k</i>) boundary edges. The result also applies to symmetric diagonally dominant matrices of planar structure.

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