A Graph-Theoretic Game and its Application to the $k$-Server Problem
SIAM Journal on Computing
Iterative solution methods
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Improved bounds for the unsplittable flow problem
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Minimizing Congestion in General Networks
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Performance evaluation of a new parallel preconditioner
IPPS '95 Proceedings of the 9th International Symposium on Parallel Processing
A practical algorithm for constructing oblivious routing schemes
Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures
A polynomial-time tree decomposition to minimize congestion
Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time 0(m1.31)
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Support Theory for Preconditioning
SIAM Journal on Matrix Analysis and Applications
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Lower bounds for graph embeddings and combinatorial preconditioners
Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
SIAM Journal on Matrix Analysis and Applications
New lower bounds for oblivious routing in undirected graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A linear work, O(n1/6) time, parallel algorithm for solving planar Laplacians
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Oblivious routing on node-capacitated and directed graphs
ACM Transactions on Algorithms (TALG)
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
Survey on Oblivious Routing Strategies
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
The laplacian paradigm: emerging algorithms for massive graphs
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
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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 x 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 off-diagonal 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 vertices i and j with weight c((i,j)) = --Ai,j = --Aj,i if Ai,j = Aj,i ‹ 0. Gremban et al showed experimentally that STCG performs well on several classes of graphs commonly used in scientific computations. In his thesis, Gremban also proved upper bounds on the number of iterations required for STCG to converge for certain classes of graphs. In this paper, we present an algorithm for finding a preconditioner for an arbitrary graph G = (V,E) with n vertices, m edges, and a weight function c › 0 on the edges, where w.l.o.g., mine∈e c(e) = 1. Equipped with this preconditioner, STCG requires O(log4 n · √Δ/α) iterations, where α = minU⊂V,|U|≤|V|/2 c(U,V\U)/|U| is the minimum edge expansion of the graph, and Δ = maxυ∈V c(υ) 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).