Foundations and Trends® in Theoretical Computer Science > Vol 9 > Issue 2

Faster Algorithms via Approximation Theory

Sushant Sachdeva, Yale University, USA, sushant.sachdeva@yale.edu Nisheeth K. Vishnoi, Microsoft Research, India, nisheeth.vishnoi@gmail.com
 
Suggested Citation
Sushant Sachdeva and Nisheeth K. Vishnoi (2014), "Faster Algorithms via Approximation Theory", Foundations and Trends® in Theoretical Computer Science: Vol. 9: No. 2, pp 125-210. http://dx.doi.org/10.1561/0400000065

Published: 28 Mar 2014
© 2014 S. Sachdeva and N. K. Vishnoi
 
Subjects
Design and analysis of algorithms,  Computational aspects of combinatorics and graph theory,  Spectral methods
 

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In this article:
1. Uniform Approximations
2. Chebyshev Polynomials
3. Approximating Monomials
4. Approximating the Exponential
5. Lower Bounds for Polynomial Approximations
6. Approximating the Exponential using Rational Functions
7. Rational Approximations to the Exponential with Negative Poles
8. Simulating Random Walks
9. Solving Linear Equations via the Conjugate Gradient Method
10. Computing Eigenvalues via the Lanczos Method
11. Computing the Matrix Exponential
12. Matrix Inversion via Exponentiation
References

Abstract

This monograph presents techniques to approximate real functions such as x s ; x–1 and e x by simpler functions and shows how these results can be used for the design of fast algorithms. The key lies in the fact that such results imply faster ways to approximate primitives such as A s v; A–1v and exp(–A)v, and to compute matrix eigenvalues and eigenvectors. Indeed, many fast algorithms reduce to the computation of such primitives, which have proved useful for speeding up several fundamental computations such as random walk simulation, graph partitioning and solving linear systems of equations.

DOI:10.1561/0400000065
ISBN: 978-1-60198-820-1
106 pp. $75.00
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ISBN: 978-1-60198-821-8
106 pp. $120.00
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Table of contents:
1. Uniform Approximations
2. Chebyshev Polynomials
3. Approximating Monomials
4. Approximating the Exponential
5. Lower Bounds for Polynomial Approximations
6. Approximating the Exponential using Rational Functions
7. Rational Approximations to the Exponential with Negative Poles
8. Simulating Random Walks
9. Solving Linear Equations via the Conjugate Gradient Method
10. Computing Eigenvalues via the Lanczos Method
11. Computing the Matrix Exponential
12. Matrix Inversion via Exponentiation
References

Faster Algorithms via Approximation Theory

Faster Algorithms via Approximation Theory illustrates how classical and modern techniques from approximation theory play a crucial role in obtaining results that are relevant to the emerging theory of fast algorithms. The key lies in the fact that such results imply faster ways to approximate primitives such as products of matrix functions with vectors, and to compute matrix eigenvalues and eigenvectors, which are fundamental to many spectral algorithms.

The first half of the book is devoted to the ideas and results from approximation theory that are central, elegant, and may have wider applicability in theoretical computer science. These include not only techniques relating to polynomial approximations but also those relating to approximations by rational functions and beyond. The remaining half illustrates a variety of ways that these results can be used to design fast algorithms.

Faster Algorithms via Approximation Theory is self-contained and should be of interest to researchers and students in theoretical computer science, numerical linear algebra, and related areas.

 
TCS-065