Advertisement

The diameter of connected components of random graphs

  • Paul Spirakis
Randomness Considerations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 246)

Abstract

This work investigates the probability distribution of the maximum of the diameters of the connected components of a random graph of the Gn,p model. (We call this maximum the depth d of the graph Gn,p). D is also defined as the maximum over all u,vεV of the quantities d(u,v) and 1, where d(u,v) is the length of the shortest path from u to v (if any) and +∞ otherwise. We prove that (1) there is a constant c>2 such that, for any probability p in the range \([0,{\text{ }}1] - [\frac{c}{n},{\text{ }}\frac{{2c}}{n} - (\frac{c}{n})^2 ]\), the graph Gn,p has average depth ↔d=0 (logn). Furthermore, the probability that d=0 (logn) tends to 1 as n tends to ∞. We also prove that for \(p \geqslant \frac{c}{{\sqrt[3]{n}}}\) (where c>1 is a particular constant) the depth of Gn,p is less than or equal to 3 with probability tending to 1 as n tends to ∞. Although the results \(\bar d = 0\left( {\log n} \right)\) can be deduced from results of [Erdös, Renyi, 60] for several values of p, the result for sparse graphs \((p = \theta (\frac{1}{n}))\) and for very dense graphs \((p \geqslant \frac{c}{{\sqrt[3]{n}}})\) are entirely new.

Keywords

Random Graph Sparse Graph Dense Graph Balance Graph Biconnected Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chin, F., J. Lam, and T. Chen, "Optimal Parallel Algorithms for the Connected Components Problem", CACM 25 (9), Sept. 1982, pp. 659–665.Google Scholar
  2. Chernoff, H., "A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations", Annals of Math. Statistics, 23 1952.Google Scholar
  3. Cook, S., "Towards a Complexity Theory of Synchronous Parallel Computations", Specker Symposium on Logic and Algorithms, Zurich, Feb. 5–11, 1980.Google Scholar
  4. Dekel, E., D. Nassimi, and S. Sahni, "Parallel Matrix and Graph Algorithms", SIAM J. Comp. 10 (4), Nov. 1981, pp. 657–675.CrossRefGoogle Scholar
  5. Dymond, P. and S.A. Cook, "Hardware Complexity and Parallel Computation", IEEE 21st Symposium on Foundations of Computer Science, 1980, pp. 360–372.Google Scholar
  6. Erdos, P. and A. Renyi, "On the Evolution of Random Graphs", Pub. Math. Inst. Hung. Acad. Sci. 5A, 1960, pp. 17–61, also "The Art of Counting", J. Spenser Editor, MIT Press.Google Scholar
  7. Feller W., "An Introduction to Probability Theory and Its Applications", Vol. 1, Third Edition, John Wiley and Sons, New York, 1968.Google Scholar
  8. Fortune, S. and J. Wyllie, "Parallelism in Random Access Machines", Proc. 10th ACM Symp. on Theory of Computing, May, 1978, pp. 114–118.Google Scholar
  9. Hirschberg, D., A. Chandra, and D. Sarwate, "Computing Connected Components on Parallel Computers", Commun. of the ACM 22 (8), Aug. 1979, pp. 461–464.Google Scholar
  10. Ja'Ja' J, "Graph Connectivity Problems on Parallel Computers", TR GS-78-05, Dept. of Computer Science, Penn. State Univ., PA, 1976.Google Scholar
  11. Nath, D. and S.N. Maheshwari, "Parallel Algorithms for the Connected Components and Minimal Spanning Tree Problems", Information Processing Letters, 14 (1), April 1982, pp. 7–11.CrossRefGoogle Scholar
  12. Reif, J. and P. Spirakis, "K-Connectivity in Random Undirected Graphs, "to appear in Journal of Discrete Mathematics, 1986.Google Scholar
  13. Savage, C. and J. Ja'Ja', "Fast, Efficient Parallel Algorithms for Some Graph Problems", SIAM J. Comp., 10 (4), Nov. 1981, pp. 682–691.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Paul Spirakis
    • 1
    • 2
  1. 1.Courant Institute of Mathematical Sciences, NYUUSA
  2. 2.Computer Technology InstituteGreece

Personalised recommendations