## A linear work, O(n^1/6) time, parallel algorithm for solving planar Laplacians

Citations: | 2 - 1 self |

### BibTeX

@MISC{Koutis_alinear,

author = {Ioannis Koutis and Gary L. Miller},

title = { A linear work, O(n^1/6) time, parallel algorithm for solving planar Laplacians },

year = {}

}

### OpenURL

### Abstract

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 ¯x such that ||x − ¯x||A ≤ ɛ, in O(n 1/6+c log(1/ɛ)) parallel time, doing O(n log(1/ɛ)) work, where c is any positive constant. One of the key ingredients of the solver, is an O(nk log 2 k) work, O(k log n) 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.