A Fast Minimum Degree Algorithm and Matching Lower Bound
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The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has rich history of bridging techniques from data structures, graph algorithms, and direct methods in scientific computing. We present a simple but novel combinatorial algorithm for computing minimum degree elimination orderings in O(nm) time that relies on a careful amortized analysis. Furthermore, we show that there cannot exist an algorithm for this problem that runs in O(nm^1-ε) time, for any ε > 0, assuming the strong exponential time hypothesis.
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