Skip to main content
SpringerLink
Log in

The Laplacian Paradigm: Emerging Algorithms for Massive Graphs

  • Conference paper
Theory and Applications of Models of Computation (TAMC 2010)

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

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abraham, I., Bartal, Y., Neiman, O.: Nearly Tight Low Stretch Spanning Trees. In: FOCS 2008, pp. 781–790 (2008)

    Google Scholar 

  2. Alon, N., Karp, R.M., Peleg, D., West, D.: A graph-theoretic game and its application to the k-server problem. SIAM Journal on Computing 24(1), 78–100 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alpert, C.J., Yao, S.-Z.: Spectral partitioning: the more eigenvectors, the better. In: DAC 1995: Proceedings of the 32nd ACM/IEEE conference on Design automation, pp. 195–200. ACM, New York (1995)

    Chapter  Google Scholar 

  4. Andersen, R., Borgs, C., Chayes, J.T., Hopcroft, J.E., Jain, K., Mirrokni, V.S., Teng, S.-H.: Robust PageRank and locally computable spam detection features. In: Fourth International Workshop on Adversarial Information Retrieval on the Web. ACM International Conference Proceeding Series, pp. 69–76 (2008)

    Google Scholar 

  5. Andersen, R., Borgs, C., Chayes, J.T., Hopcraft, J.E., Mirrokni, V.S., Teng, S.-H.: Local computation of pagerank contributions. In: Bonato, A., Chung, F.R.K. (eds.) WAW 2007. LNCS, vol. 4863, pp. 150–165. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Andersen, R., Chung, F., Lang, K.: Local graph partitioning using pagerank vectors. In: Proceedings: 47th Annual Symposium on Foundations of Computer Science, pp. 475–486 (2006)

    Google Scholar 

  7. Andersen, R., Peres, Y.: Finding sparse cuts locally using evolving sets. In: STOC, pp. 235–244 (2009)

    Google Scholar 

  8. Arora, S., Hazan, E., Kale, S.: 0(sqrt (log n)) approximation to Sparsest Cut in õ(n2) time. In: FOCS: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 238–247 (2004)

    Google Scholar 

  9. Arora, S., Kale, S.: A combinatorial, primal-dual approach to semidefinite programs. In: STOC 2007: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pp. 227–236. ACM, New York (2007)

    Chapter  Google Scholar 

  10. Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. In: STOC 2004: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pp. 222–231. ACM, New York (2004)

    Chapter  Google Scholar 

  11. Batson, J., Spielman, D.A., Srivastava, N.: Twice-Ramanujan sparsifiers (2008), http://arxiv.org/abs/0808.0163

  12. Bebendorf, M., Hackbusch, W.: Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with L  ∞ -coefficients. Numerische Mathematik 95(1), 1–28 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Benczúr, A.A., Karger, D.R.: Approximating s-t minimum cuts in O(n2) time. In: Proceedings of The Twenty-Eighth Annual ACM Symposium on the Theory of Computing, STOC 1996 (1996)

    Google Scholar 

  14. Bern, M., Gilbert, J., Hendrickson, B., Nguyen, N., Toledo, S.: Support-graph preconditioners. SIAM J. Matrix Anal. and Appl. 27(4), 930–951 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Boman, E.G., Hendrickson, B.: Support theory for preconditioning. SIAM Journal on Matrix Analysis and Applications 25(3), 694–717 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Boman, E.G., Hendrickson, B., Vavasis, S.: Solving epplitic finite element systems in nearly-linear time with support preconditioners

    Google Scholar 

  17. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2001)

    Google Scholar 

  18. Cheeger, J.: A lower bound for smallest eigenvalue of laplacian. In: Gunning, R.C. (ed.) Problems in Analysis, pp. 195–199. Princeton University Press, Princeton (1970)

    Google Scholar 

  19. Chew, P.: There is a planar graph almost as good as the complete graph. In: SCG 1986: Proceedings of the second annual symposium on Computational geometry, pp. 169–177. ACM, New York (1986)

    Chapter  Google Scholar 

  20. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn.

    Google Scholar 

  21. Daitch, S.I., Spielman, D.A.: Support-graph preconditioners for 2-dimensional trusses

    Google Scholar 

  22. Daitch, S.I., Spielman, D.A.: Faster approximate lossy generalized flow via interior point algorithms. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 451–460 (2008)

    Google Scholar 

  23. Elkin, M., Emek, Y., Spielman, D.A., Teng, S.-H.: Lower-stretch spanning trees. SIAM Journal on Computing 32(2), 608–628 (2008)

    Article  MathSciNet  Google Scholar 

  24. Frieze, A.M., Miller, G.L., Teng, S.-H.: Separator Based Parallel Divide and Conquer in Computational Geometry. In: SPAA 1992, pp. 420–429 (1992)

    Google Scholar 

  25. George, J.A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10, 345–363 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gilbert, J.R., Tarjan, R.E.: The analysis of a nested dissection algorithm. Numerische Mathematik 50(4), 377–404 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  27. Golub, G.H., Overton, M.: The convergence of inexact Chebychev and Richardson iterative methods for solving linear systems. Numerische Mathematik 53, 571–594 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gremban, K.: Combinatorial Preconditioners for Sparse, Symmetric, Diagonall y Dominant Linear Systems. PhD thesis, Carnegie Mellon University, CMU-CS-96-123 (1996)

    Google Scholar 

  29. Gulli, A., Signorini, A.: The indexable web is more than 11.5 billion pages. In: WWW 2005: Special interest tracks and posters of the 14th international conference on World Wide Web, pp. 902–903. ACM, New York (2005)

    Chapter  Google Scholar 

  30. Joshi, A.: Topics in Optimization and Sparse Linear Systems, Ph.D. thesis, UIUC (1997)

    Google Scholar 

  31. Khandekar, R., Rao, S., Vazirani, U.: Graph partitioning using single commodity flows. In: STOC 2006: Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pp. 385–390. ACM, New York (2006)

    Chapter  Google Scholar 

  32. Kolla, A., Makarychev, Y., Saberi, A., Teng, S.-H.: Subgraph Sparsification. In: STOC 2010 (2010)

    Google Scholar 

  33. Koutis, I., Miller, G., Peng, R.: Approaching optimality for solving SDD systems

    Google Scholar 

  34. Koutis, I., Miller, G., Tolliver, D.: Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing. In: International Symp. of Visual Computing, pp. 1067–1078 (2009)

    Google Scholar 

  35. Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM Journal on Numerical Analysis 16(2), 346–358 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  36. Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46(6), 787–832 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lovasz, L., Simonovits, M.: Random walks in a convex body and an improved volume algorithm. RSA: Random Structures & Algorithms 4, 359–412 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  38. Maggs, B., Miller, G., Parekh, O., Ravi, R., Woo, S.M.: Finding effective support-tree preconditioners. In: ACM SPAA, pp. 176–185 (2005)

    Google Scholar 

  39. Madry, A., Kelner, J.: Faster generation of random spanning trees. In: FOCS (2009)

    Google Scholar 

  40. Miller, G.L., Teng, S.-H., Thurston, W.P., Vavasis, S.A.: Separators for sphere-packings and nearest neighbor graphs. J. ACM 44(1), 1–29 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  41. Orecchia, L., Schulman, L.J., Vazirani, U.V., Vishnoi, N.K.: On partitioning graphs via single commodity flows. In: STOC 2008: Proceedings of the 40th annual ACM symposium on Theory of computing, pp. 461–470. ACM, New York (2008)

    Chapter  Google Scholar 

  42. Rief, J.: Efficient approximate solution of sparse linear systems. Computer and Mathematics with Applications 36(9), 37–58 (1998)

    Article  Google Scholar 

  43. Sharma, A., Liu, X., Miller, P., Nakano, A., Kalia, R.K., Vashishta, P., Zhao, W., Campbell, T.J., Haas, A.: Immersive and interactive exploration of billion-atom systems. In: VR 2002: Proceedings of the IEEE Virtual Reality Conference 2002, p. 217. IEEE Computer Society, Los Alamitos (2002)

    Chapter  Google Scholar 

  44. Shklarski, G., Toledo, S.: Rigidity in finite-element matrices: Sufficient conditions for the rigidity of structures and substructures. SIAM J. Matrix Anal. and Appl. 30(1), 7–40 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  45. Spielman, D.: Graphs and networks: Random walks and spectral graph drawing. Computer Science, Yale (September 18, 2007), http://www.cs.yale.edu/homes/spielman/462/lect4-07.pdf

  46. Spielman, D.A., Srivastava, N.: Graph sparsification by effective resistances. In: Proceedings of the 40th annual ACM Symposium on Theory of Computing, pp. 563–568 (2008)

    Google Scholar 

  47. Spielman, D., Teng, S.-H.: Spectral partitioning works: planar graphs and finite element meshes. In: FOCS 1996: Proceedings of the 37th annual symposium on Foundations of Computer Science, pp. 96–105 (1996)

    Google Scholar 

  48. Spielman, D.A., Teng, S.-H.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pp. 81–90 (2003)

    Google Scholar 

  49. Spielman, D.A., Teng, S.-H.: A local clustering algorithm for massive graphs and its application to nearly-linear time graph partitioning. CoRR, abs/0809.3232 (2008), http://arxiv.org/abs/0809.3232

  50. Spielman, D.A., Teng, S.-H.: Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. CoRR, abs/cs/0607105 (2008), http://www.arxiv.org/abs/cs.NA/0607105

  51. Spielman, D.A., Teng, S.-H.: Spectral sparsification of graphs. CoRR, abs/0808.4134 (2008), http://arxiv.org/abs/0808.4134

  52. Tolliver, D.A.: Spectral rounding and image segmentation. PhD thesis, Pittsburgh, PA, USA (2006); Adviser-Miller, G.L., Adviser-Collins, R.T.

    Google Scholar 

  53. Tolliver, D.A., Miller, G.L.: Graph partitioning by spectral rounding: Applications in image segmentation and clustering. In: CVPR 2006: Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1053–1060. IEEE Computer Society, Los Alamitos (2006)

    Google Scholar 

  54. Trefethen, L.N., Bau, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  55. Vaidya, P.: Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners. An invited at IMA, U. Minnesota (October 1991)

    Google Scholar 

  56. Zhou, D., Huang, J., Schölkopf, B.: Learning from Labeled and Unlabeled Data on a Directed Graph. In: The 22nd International Conference on Machine Learning, pp. 1041–1048 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, University of Southern California,  

    Shang-Hua Teng

Authors
  1. Shang-Hua Teng
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  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

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

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

Download citation

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

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13561-3

  • Online ISBN: 978-3-642-13562-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation