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The SIAM series on Software, Environments, and Tools focuses on the practical implementation of computational methods and the high performance aspects of scientific computation by emphasizing in-demand software, computing environments, and tools for computing. Software technology development issues such as current status, applications and algorithms, mathematical software, software tools, languages and compilers, computing environments, and visualization are presented. Editor-in-Chief Jack J. Dongarra University of Tennessee and Oak Ridge National Laboratory Editorial Board James W. Demmel, University of California, Berkeley Dennis Gannon, Indiana University Eric Grosse, AT&T Bell Laboratories Jorge J. Moré, Argonne National Laboratory ´ Retrieval, Second Edition and Students P. Yalamov, LAPACK95 Users’ Guide Software, Environments, and Tools Jeremy Kepner and John Gilbert, editors, Graph Algorithms in the Language of Linear Algebra Jeremy Kepner, Parallel MATLAB for Multicore and Multinode Computers Michael A. Heroux, Padma Raghavan, and Horst D. Simon, editors, Parallel Processing for Scientific Computing Gérard Meurant, The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations Bo Einarsson, editor, Accuracy and Reliability in Scientific Computing Michael W. Berry and Murray Browne, Understanding Search Engines: Mathematical Modeling and Text Craig C. Douglas, Gundolf Haase, and Ulrich Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization Louis Komzsik, The Lanczos Method: Evolution and Application Bard Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers V. A. Barker, L. S. Blackford, J. Dongarra, J. Du Croz, S. Hammarling, M. Marinova, J. Wasniewski, and Stefan Goedecker and Adolfy Hoisie, Performance Optimization of Numerically Intensive Codes Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst, Templates for the Solution Lloyd N. Trefethen, Spectral Methods in MATLAB E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, Third Edition Michael W. Berry and Murray Browne, Understanding Search Engines: Mathematical Modeling and Text Retrieval Jack J. Dongarra, Iain S. Duff, Danny C. Sorensen, and Henk A. van der Vorst, Numerical Linear Algebra R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods Randolph E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users’ Guide 8.0 L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide Greg Astfalk, editor, Applications on Advanced Architecture Computers Roger W. Hockney, The Science of Computer Benchmarking Françoise Chaitin-Chatelin and Valérie Frayssé, Lectures on Finite Precision Computations of Algebraic Eigenvalue Problems: A Practical Guide for High-Performance Computers SE22_Kepner_FM-04-28-11.indd 2 Downloaded 09 Dec 2011 to 129.174.55.245. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 5/12/2011 10:14:49 AM

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Copyright © 2011 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, info@mathworks.com, www.mathworks.com. This work is sponsored by the Department of the Air Force under Air Force Contract FA8721- 05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. Library of Congress Cataloging-in-Publication Data Kepner, Jeremy V., 1969- Graph algorithms in the language of linear algebra / Jeremy Kepner, John Gilbert. p. cm. -- (Software, environments, and tools) Includes bibliographical references and index. ISBN 978-0-898719-90-1 1. Graph algorithms. 2. Algebras, Linear. I. Gilbert, J. R. (John R.), 1953- II. Title. QA166.245.K47 2011 511’.6--dc22 2011003774 is a registered trademark. SE22_Kepner_FM-04-28-11.indd 4 Downloaded 09 Dec 2011 to 129.174.55.245. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 5/12/2011 10:14:49 AM

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List of Contributors David A. Bader College of Computing Georgia Institute of Technology Atlanta, GA 30332 bader@cc.gatech.edu Nadya Bliss MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 nt@ll.mit.edu Robert Bond MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 rbond@ll.mit.edu Aydın Buluç High Performance Computing Research Lawrence Berkeley National 1 Cyclotron Road Berkeley, CA 94720 abuluc@lbl.gov Laboratory John Gilbert Computer Science Department University of California at Santa Barbara Santa Barbara, CA 93106 gilbert@cs.ucsb.edu Christine E. Heitsch School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 heitsch@math.gatech.edu Bruce Hendrickson Discrete Algorithms and Mathematics Department Sandia National Laboratories Albuquerque, NM 87185 bahendr@sandia.gov Sanjeev Mohindra MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 smohindra@ll.mit.edu Huy Nguyen MIT Computer Science and Artificial Intelligence Laboratory 32 Vassar Street Cambridge, MA 02139 huy2n@mit.edu (CSAIL) Charles M. Rader MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 charlesmrader@verizon.net W. Philip Kegelmeyer Informatics and Decision Sciences Department Sandia National Laboratories Livermore, CA 94551 wpk@sandia.gov Steve Reinhardt Microsoft Corporation 716 Bridle Ridge Road Eagan, MN 55123 steve.reinhardt@microsoft.com Eric Robinson MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 erobinson@ll.mit.edu Viral B. Shah 82 E Marine Drive Badrikeshwar, Flat No. 25 Mumbai 400 002 India viral@mayin.org Daniel M. Dunlavy Computer Science and Informatics Department Sandia National Laboratories Albuquerque, NM 87185 dmdunla@sandia.gov Jeremy Kepner MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 kepner@ll.mit.edu Alan Edelman Mathematics Department MIT 77 Massachusetts Avenue Cambridge, MA 02139 edelman@mit.edu Christos Faloutsos School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 christos@cs.cmu.edu Jeremy T. Fineman School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 jfineman@cs.cmu.edu Tamara G. Kolda Informatics and Decision Sciences Department Sandia National Laboratories Livermore, CA 94551 tgkolda@sandia.gov Jure Leskovec Computer Science Department Stanford University Stanford, CA 94305 jure@cs.stanford.edu Kamesh Madduri Computational Research Division Lawrence Berkeley National Berkeley, CA 94720 kmadduri@lbl.gov Laboratory vii SE22_Kepner_FM-04-28-11.indd 6 Downloaded 09 Dec 2011 to 129.174.55.245. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 5/12/2011 10:14:50 AM

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