...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
template <class AdjacencyGraph, class OutDegreeMap, class InversePermutationMap, class PermutationMap, class SuperNodeSizeMap, class VertexIndexMap> void minimum_degree_ordering (AdjacencyGraph& G, OutDegreeMap outdegree, InversePermutationMap inverse_perm, PermutationMap perm, SuperNodeSizeMap supernode_size, int delta, VertexIndexMap id)
The minimum degree ordering algorithm [21, 35] is a fill-in reduction matrix reordering algorithm. When solving a system of equations such as A x = b using a sparse version of Cholesky factorization (which is a variant of Gaussian elimination for symmetric matrices), often times elements in the matrix that were once zero become non-zero due to the elimination process. This is what is referred to as "fill-in", and fill-in is bad because it makes the matrix less sparse and therefore consume more time and space in later stages of the elimintation and in computations that use the resulting factorization. Now it turns out that reordering the rows of a matrix can have a dramatic affect on the amount of fill-in that occurs. So instead of solving A x = b, we instead solve the reordered (but equivalent) system (P A P^{T})(P x) = P b. Finding the optimal ordering is an NP-complete problem (e.i., it can not be solved in a reasonable amount of time) so instead we just find an ordering that is "good enough" using heuristics. The minimum degree ordering algorithm is one such approach.
A symmetric matrix A is typically represented with an undirected graph, however for this function we require the input to be a directed graph. Each nonzero matrix entry A(i, j) must be represented by two directed edges (e(i,j) and e(j,i)) in G.
The output of the algorithm are the vertices in the new ordering.
inverse_perm[new_index[u]] == old_index[u]
and the permutation from the old index to the new index.
perm[old_index[u]] == new_index[u]
The following equations are always held.
for (size_type i = 0; i != inverse_perm.size(); ++i) perm[inverse_perm[i]] == i;
UNDER CONSTRUCTION
It chooses the vertex with minimum degree in the graph at each step of simulating Gaussian elimination process.
This implementation provided a number of enhancements including mass elimination, incomplete degree update, multiple elimination, and external degree. See [35] for a historical survey of the minimum degree algorithm.
An elimination graph G_{v} is obtained from the
graph G by removing vertex v and all of its incident
edges and by then adding edges to make all of the vertices adjacent to
v into a clique (that is, add an edge between each pair of
adjacent vertices if an edge doesn't already exist).
Suppose that graph G is the graph representing the nonzero
structure of a matrix A. Then performing a step of Guassian
elimination on row i of matrix A is equivalent to
Copyright © 2001 |
Lie-Quan Lee, Indiana University (llee@cs.indiana.edu) Jeremy Siek, Indiana University (jsiek@osl.iu.edu) |