Lecture 1: Low-stretch trees
نویسنده
چکیده
The main theme of the workshop is fast algorithms, particularly those that relate to fast solvers for linear systems involving the Laplacian of a graph. In my lectures, I will discuss three key technical ingredients that underlie those solvers. In this first lecture, I will discuss “low-stretch trees”. Given a graph, the goal is to find a spanning subtree such that, on average, distances in the tree approximate distances in the graph. These trees have many uses, but we will eventually see that they are particularly useful as preconditioners of Laplacian matrices.
منابع مشابه
A Note on Preconditioning by Low-Stretch Spanning Trees
Boman and Hendrickson [BH01] observed that one can solve linear systems in Laplacian matrices in time O ( m ln(1/ǫ) ) by preconditioning with the Laplacian of a low-stretch spanning tree. By examining the distribution of eigenvalues of the preconditioned linear system, we prove that the preconditioned conjugate gradient will actually solve the linear system in time Õ ( m ln(1/ǫ) ) .
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