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Minimum-Degree Ordering

Minimum-degree ordering has been used in our research in a two-fold manner:

  1. to order symmetric power system admittance matrices to provide baseline orderings with which to compare the performance of other ordering techniques
  2. to order the independent sub-matrices in recursive spectral bisection and node-tearing ordering techniques
Minimum degree ordering is a greedy algorithm that selects a node with a minimum number of connected edges in the graph for factoring next. This algorithm is not optimal because truly efficient techniques do not exists to resolve ties and numerous rows have equal numbers of elements. The minimum-degree ordering algorithm is based on the iterative application of the following equation to solve for i for all rows in a matrix:

 where:¯

is the number of variables in row i when factoring the row.

is the number of variables in row t when factoring the row

When factoring the row, the row with the minimum number of variables is selected, moved by elementary row and column exchange rules to the row, and then factored. Algorithms to implement this iterative formula are best described using the graph theoretical explanation of fillin presented in figure 9. Let G be an undirected graph and a node in G, then let describe the set of nodes adjacent to and let represent the degree of node . The last concept required to develop a concise minimum-degree algorithm is the concept of an elimination graph [17]. Given a graph G, the elimination graph is the resulting graph after the node is factored. Elimination graphs get their name because of the close relationship of LU factorization and Gaussian elimination. The rudimentary minimum-degree algorithm used throughout this work is presented in figure 63. The outer loop examines each node in the graph, and the inner loop searches through all remaining nodes in the present graph to select a node with the minimum degree. After a minimum-degree node is selected, the edges at adjacent nodes must be updated to reflect factorization. As illustrated in figure 9, the addition of new edges in the elimination graph is limited to those nodes in . For , then

 
Figure 63: The Minimum-Degree Algorithm 

Given the two nested loops that can examine all nodes in the original sparse graph, the computational order of this algorithm is , although a significant portion of the workload is required to calculate the elimination graph [17]. As stated above, in formula 4, the total amount of calculations in the loop to update the elimination graph is bounded by the binomial coefficient of the number of edges at a node choose 2 or chose 2. See equation 4 for details on calculating the binomial coefficient. It is important to note that the location of all fillin can be determined when using this classical implementation of minimum degree ordering.

This version of the minimum-degree algorithm has been used in our research in a two-fold manner: to order symmetric power systems admittance matrices to provide baseline orderings with which to compare the performance of other ordering techniques, and to order the independent sub-matrices obtained with node-tearing ordering techniques.



next up previous contents
Next: A Node-tearing Example Up: Parallel Direct Methods for Previous: References



David P. Koester
Sun Oct 22 15:31:10 EDT 1995