Finding effective support-tree preconditioners
Bruce MacDowell Maggs, Gary Lee Miller, Ojas Parekh, R. Ravi, Shan Leung Maverick Woo · 2005
In 1995, Gremban, Miller, and Zagha introduced support-tree preconditioners and a parallel algorithm called support-tree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the off-diagonal entries are non-positive, and the row and column sums are zero. A Laplacian A with 2m non-zeros can be interpreted as an undirected positively-weighted graph G with n vertices and m edges, where there is an edge between two nodes i and j with weight c((i, j)) = −Ai,j = −Aj,i if Ai,j = Aj,i 0 on the edges, where w.l.o.g., mine∈E c(e) = 1. Equipped with this preconditioner, STCG requires O(log 4 n · � ∆/α) iterations, where α = min U⊂V,|U|≤|V |/2 c(U, V \\U)/|U | is the minimum edge expansion of the graph, and ∆ = maxv∈V c(v) is the maximum incident weight on any vertex. Each iteration requires O(m) work and can be implemented in O(log n) steps in parallel, using only O(m) space. Our results generalize to matrices that are symmetric and diagonally-dominant (SDD). 1