Parallelizing Elimination Orders with Linear Fill

This paper presents an algorithm for nding parallel elimination orders for Gaussian elimination Viewing a system of equations as a graph the algorithm can be applied directly to interval graphs and chordal graphs For general graphs the algorithm can be used to paral lelize the order produced by some other heuristic such as minimum degree In this case the algorithm is ap plied to the chordal completion that the heuristic gen erates from the input graph In general the input to the algorithm is a chordal graph G with n nodes and m edges The algorithm produces an order with height at most O log n times optimal ll at most O m and work at most O W G where W G is the minimum possible work over all elimination orders for G Experimental results show that when applied after some other heuristic the increase in work and ll is usually small In some instances the algorithm obtains an order that is actually better in terms of work and ll than the original one We also present an algo rithm that produces an order with a factor of logn less height but with a factor of O p logn more ll