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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. 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 5. 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
[12]. 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 32. 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 5, the addition of new edges in the elimination
graph
is limited to those nodes in
. For
, then

Figure 32: 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
[12]. 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: A Node-tearing Example
Up: Parallel Choleski Factorization of
Previous: References
David P. Koester
Sun Oct 22 15:40:25 EDT 1995