Next: Recursive Spectral Bisection-Based
Up: Ordering Sparse Matrices
Previous: Ordering Sparse Matrices
Minimum-degree ordering has been
used in our research in a two-fold manner:
- to order symmetric power system admittance matrices to provide baseline orderings with which to compare the performance of other ordering techniques
- 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,
or the row to be factored next is that row with a minimum number of
variables [5]. This algorithm is not optimal because
truly efficient techniques do not exist to resolve ties and numerous
rows have equal numbers of elements. Various versions of minimum
degree ordering exist, with some versions employing heuristics to
minimize the number of calculations. Examples of recursive spectral
bisection-based ordering and node-tearing-based ordering are presented
below. To contrast the performance of those ordering techniques,
figure 3 illustrates a minimum degree ordering of the
BCSPWR09.PSA matrix from the Boeing-Harwell series [4]. This
matrix represents an actual 1723 bus western US power network, and
will be used for all ordering comparisons. Note that the matrix is
the most sparse in the upper left-hand corner, while the matrix is
less sparse in the lower right-hand corner. When factoring this
matrix, the number of zero values that become non-zero while factoring
the matrix, is 2,168. Original nonzero values are represented in this
figure in black, fillin locations are represented in gray, and all
remaining zero values are white. A bounding box has been placed
around the sparse matrix.
Figure 3: Minimum Degree Ordering
David P. Koester
Sun Oct 22 17:45:03 EDT 1995