Abstract
We consider the task of evaluating properties of graphs that are too big to be even scanned. Thus, the input graph is given in form of an oracle which answers questions of the form is there an edge between vertices u and v, or who is the ith neighbor of v. Our task is to determine whether a given input graph has a predetermined property or is “relatively far” from any graph having the property. Distance between graphs is measured as the fraction of the possible queries on which the corresponding oracles, representing the two graphs, differ. We show that randomized algorithms of running-time substantially smaller than the size of the input graph may reach (with high probability) a correct verdict regarding whether the graph has some predetermined property (such as being bipartite) or is far from having it.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy. Efficient testing of large graphs. In Proceedings of the 40th IEEE Symp on Foundation of Computer Science, pages 656–666, 1999.
N. Alon and M. Krivelevich. Testing k-colorability, 1999. Preprint.
N. Alon and J.H. Spencer. The Probabilistic Method. John Wiley & Sons, Inc., 1992.
S. Arora, D. Karger, and M Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, pages 284–293, 1995.
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. Journal of the ACM, 45: 501–555, 1998.
S. Arora and S. Safra. Probabilistic checkable proofs: A new characterization of NP. Journal of the ACM, 45: 70–122, 1998.
L. Babai, L. Fortnow, L. Levin, and M. Szegedy. Checking computations in polylogarithmic time. In Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pages 21–31, 1991a.
L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1: 3–40, 1991b.
M. Bellare, D. Coppersmith, J. Hâ,stad, M. Kiwi, and M. Sudan. Linearity testing in characteristic two. IEEE Transactions on Information Theory, 42: 1781–1795, 1996.
M. Bellare, O. Goldreich, and M. Sudan. Free bits, pcps and nonapproximability–Towards tight results. SICOMP, 27: 804–915, 1998.
M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the Twenty-Fifth Annual ACM Symposium on the Theory of Computing, pages 294–304, 1993.
M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pages 184–193, 1994.
M. Bender and D. Ron. Testing acyclicity of directed graphs in sublinear time. In Proceedings of ICALP, pages 809–820, 2000.
M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47: 549–595, 1993.
W. F. de la Vega. MAX-CUT has a randomized approximation scheme in dense graphs. Random Structures and Algorithms, 1994. To appear.
Y. Dodis, O. Goldreich, E. Lehman, S. Raskhodnikova, D. Ron, and A. Samorodnitsky. Improved testing algorithms for monotonocity. In Proceedings of Random99, pages 97–108, 1999.
F. Ergun, S. Kannan, S. R. Kumar, R. Rubinfeld, and M. Viswanathan. Spot-checkers. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 259–268, 1998.
S. Even. Graph Algorithms. Computer Science Press, 1979.
U. Feige, S. Goldwasser, L. Lovâsz, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. Journal of the ACM, 43: 268–292, 1996.
P. Gemmell, R. Lipton, R. Rubinfeld, M. Sudan, and A. Wigderson. Selftesting/correcting for polynomials and for approximate functions. In Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pages 32–42, 1991.
O. Goldreich, S. Goldwasser, E. Lehman, and D. Ron. Testing monotinicity. In Proceedings of the 39th IEEE Symp on Foundation of Computer Science, pages 426–435, 1998a.
O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. Journal of the ACM, pages 653–750, 1998b.
O. Goldreich and D. Ron. Property testing in bounded degree graphs. In Proceedings of the 29th ACM Symp. on Theory of Computing, pages 406–415, 1997.
O. Goldreich and D. Ron. A sublinear bipartite tester for bounded degree graphs. Combinatorica, 19: 1–39, 1999.
P. Hajnal. An Sì(n4/3) lower bound on the randomized complexity of graph properties. Combinatorica, 11: 131–144, 1991.
J. Hâstad. Clique is hard to approximate within n1. In Proceedings of the 37th IEEE Symp on Foundation of Computer Science, pages 627–636, 1996a.
J. Hâstad. Testing of the long code and hardness for clique. In Proceedings of the 28th ACM Symp. on Theory of Computing,pages 11–19, 19966.
J. Hâstad. Getting optimal in-approximability results. In Proceedings of the 29th ACM Symp. on Theory of Computing, pages 1–10, 1997.
D. S. Hochbaum and D. B. Shmoys. Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the Association for Computing Machinery, 34: 144–162, 1987.
D. S. Hochbaum and D. B. Shmoys. A polynomial approximation scheme for machine scheduling on uniform processors: Using the dual approximation approach. SIAM Journal on Computing, 17: 539–551, 1988.
M. Kearns and D. Ron. Testing problems with sub-learning sample complexity. In Proceedings of the 11th COLT, pages 268–277, 1998.
V. King. An S2(n514) lower bound on the randomized complexity of graph properties. Combinatorica, 11: 23–32, 1991.
M. Kiwi. Probabilistically Checkable Proofs and the Testing of Hadamard-like Codes. PhD thesis, Massachusetts Institute of Technology, 1996.
R. J. Lipton. New directions in testing, 1989. Unpublished manuscript.
L. Lovâsz and N. Young. Lecture notes on evasiveness of graph properties. Technical Report TR-317–91, Computer Science Department, Princeton University, 1991.
M. Mihail. Conductance and convergence of Markov chains — A combinatorial treatment of expanders. In Proceedings 30th Annual Symp on Foundations of Computer Science, pages 526–531, 1989.
M. Parnas and D. Ron. Testing the diameter of graphs. In Proceedings of Random99, pages 85–96, 1999.
R. L. Rivest and J. Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3: 371–384, 1976.
A. L. Rosenberg. On the time required to recognize properties of graphs: A problem. SIGACT News, 5: 15–16, 1973.
R. Rubinfeld and M. Sudan. Robust characterization of polynomials with applications to program testing. SIAM Journal on Computing, 25: 252–271, 1996.
L. Trevisan. Recycling queries in PCPs and in linearity tests. In Pro- ceedings of the 30th ACM Symp. on Theory of Computing, 1998.
L.G. Valiant. A theory of the learnable. Communications of the ACM, 27: 1134–1142, 1984.
A. C. C. Yao. Lower bounds to randomized algorithms for graph properties. In Proceedings of the Twenty-Eighth Annual Symposium on Foundations of Computer Science, pages 393–400, 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Goldreich, O. (2002). Property Testing in Massive Graphs. In: Abello, J., Pardalos, P.M., Resende, M.G.C. (eds) Handbook of Massive Data Sets. Massive Computing, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0005-6_5
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0005-6_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-4882-5
Online ISBN: 978-1-4615-0005-6
eBook Packages: Springer Book Archive