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The Laplacian Paradigm: Emerging Algorithms for Massive Graphs

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Theory and Applications of Models of Computation (TAMC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6108))

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Abstract

This presentation describes an emerging paradigm for the design of efficient algorithms for massive graphs. This paradigm, which we will refer to as the Laplacian Paradigm, is built on a recent suite of nearly-linear time primitives in spectral graph theory developed by Spielman and Teng, especially their solver for linear systems A x = b, where A is the Laplacian matrix of a weighted, undirected n-vertex graph and b is an n-place vector.

In the Laplacian Paradigm for solving a problem (on a massive graph), we reduce the optimization or computational problem to one or multiple linear algebraic problems that can be solved efficiently by applying the nearly-linear time Laplacian solver. So far, the Laplacian paradigm already has some successes. It has been applied to obtain nearly-linear-time algorithms for applications in semi-supervised learning, image process, web-spam detection, eigenvalue approximation, and for solving elliptic finite element systems. It has also been used to design faster algorithms for generalized lossy flow computation and for random sampling of spanning trees.

The goal of this presentation is to encourage more researchers to consider the use of the Laplacian Paradigm to develop faster algorithms for solving fundamental problems in combinatorial optimization (e.g., the computation of matchings, flows and cuts), in scientific computing (e.g., spectral approximation), in machine learning and data analysis (such as for web-spam detection and social network analysis), and in other applications that involve massive graphs.

Most materials in this presentation are joint work with Dan Spielman. This presentation also summarizes a recent NSF proposal of the author titled “Nearly-Linear-Time Algorithms for Massive Graphs: Spectral Graph Theory Approach.” Therefore, some discussions of this presentation are about proposed work rather than completed work. This research is supported by an NSF grant.

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Authors and Affiliations

  1. Computer Science Department, University of Southern California,  

    Shang-Hua Teng

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  1. Shang-Hua Teng
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  1. Charles University, Malostranske nam 25, 11800, Praha 1, Czech Republic

    Jan Kratochvíl , Jiří Fiala  & Petr Kolman ,  & 

  2. State Key Lab. of Computer Science, Institute of Software,, Chinese Academy of Sciences, 100190, Beijing, P.R. China

    Angsheng Li

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Teng, SH. (2010). The Laplacian Paradigm: Emerging Algorithms for Massive Graphs. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds) Theory and Applications of Models of Computation. TAMC 2010. Lecture Notes in Computer Science, vol 6108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13562-0_2

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  • DOI: https://doi.org/10.1007/978-3-642-13562-0_2

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