A Graph-Theoretic Game and its Application to the $k$-Server Problem
SIAM Journal on Computing
Topics in optimization and sparse linear systems
Topics in optimization and sparse linear systems
Steiner points in tree metrics don't (really) help
SODA '01 Proceedings of the twelfth 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
Lower Bounds for Embedding Graphs into Graphs of Smaller Characteristic
FST TCS '02 Proceedings of the 22nd Conference Kanpur on Foundations of Software Technology and Theoretical Computer Science
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
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
Bounded Geometries, Fractals, and Low-Distortion Embeddings
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
SIAM Journal on Matrix Analysis and Applications
Finding effective support-tree preconditioners
Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
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
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
Hi-index | 0.00 |
Given a general graph G, a fundamental problem is to find a spanning tree H that best approximates G by some measure. Often this measure is some combination of the congestion and dilation of an embedding of G into H. One example is the routing time p(G, H) ≤ O (congestion + dilation), the number of steps necessary to route pairwise demands G on network links H in the store-and-forward packet routing model. Another is the condition number kf(G, H) ≤ O (congestion · dilation), the square root of which bounds the number of iterations necessary to solve a linear system with coefficient matrix G preconditioned by H using the classical conjugate gradient method. The algorithmic applications of being able to find (efficiently) a good tree approximation H for a graph G are numerous; but what if no good tree exists.In this paper, we seek to identify the class of graphs G which are intrinsically difficult to approximate by a particular measure. It is easily seen that with respect to routing time, G is hardest to approximate by a tree H precisely when it contains either long cycles (which yield high dilation)or large separators (which yield high congestion). We show that with respect to condition number, the existence of long cycles or large separators in G is sufficient but not necessary or it to be hardest to approximate, by demonstrating a nearly-linear lower bound or the case in which G is a square mesh. The proof uses concepts from circuit theory, linear algebra, and geometry, and it generalizes to the case in which H is a spanning subgraph of G of Euler characteristic k. The result has consequences or the design of preconditioners or symmetric M-matrices and perhaps also of communication networks.