minimumdegree

Minimumdegree is an algorithm used in graph theory and sparse matrix computations for finding an optimal vertex ordering to minimize fill-in during elimination processes. The fill-in refers to the creation of new non-zero elements in a matrix during factorization, which can significantly increase computational complexity. The minimumdegree algorithm works by iteratively selecting the vertex with the smallest degree (number of adjacent vertices) to eliminate next. This approach tends to preserve sparsity in the matrix representation of the graph. When a vertex is eliminated, its neighbors form a clique, potentially increasing the degree of remaining vertices. The algorithm was first introduced by Alan George in 1971 and has since been refined through various implementations like minimum degree with external degree (MMD) and approximate minimum degree (AMD). These improvements address computational challenges in large-scale problems. Minimumdegree ordering is widely used in finite element analysis, circuit simulation, and other applications requiring efficient solution of large sparse linear systems. While not always producing the theoretically optimal ordering, it generally performs well in practice and remains a fundamental technique in sparse matrix computations.