Advertisement

The bisection problem for graphs of degree 4 (configuring transputer systems)

  • Juraj Hromkovič
  • Burkhard Monien
Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 520)

Abstract

It is well known that for each k≥3 there exists such a constant c k and such an infinite sequence {Gn} n=8 of k-degree graphs (each G n has exactly n vertices) that the bisection width of G n is at least c k ·n. It this paper some upper bounds on ck's are found. Let σk(n) be the maximum of bisection widths of all k-bounded graphs of n vertices. We prove that
$$\sigma _k \left( n \right) \leqslant \frac{{\left( {k - 2} \right)}}{4} \cdot n + o\left( n \right)$$
for all k=2r, r≥2. This result is improved for k=4 by constructing two algorithms A and B, where for a given 4-degree-bounded graph G n of n vertices
  1. (i)

    A constructs a bisection of G n involving at most n/2+4 edges for even n≤76 (i.e., σ4(n)≤n/2+4 for even n≤76)

     
  2. (ii)

    B constructs a bisection of G n involving at most n/2+2 edges for n≥256 (i.e. σ4(n)≤n/2+2 for n≥256).

     

The algorithms A and B run in O(n2) time on graphs of n vertices, and they are used to optimize hardware by building large transputer systems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al86.
    N. Alon: Eigenvalues and expanders. Combinatorica 6 (1986), 85–95.Google Scholar
  2. AM85.
    N. Alon — V.D. Milman: λ1, isometric inequalities for graphs, and superconcentrators. J. Combinatorial Theory B 38 (1985), 73–88.Google Scholar
  3. BCLS87.
    T.N. Bui — S. Chanduri — F.T. Leighton — M. Sipser: Graph bisection algorithms with good average case behavior. Combinatorica 7 (1987), 171–191.Google Scholar
  4. BS87.
    A. Broder — E. Shamir: On the second eigenvalue of random regular graphs. In: Proc. 28th Annual Symp. on FOCS, IEEE 1987, 286–294.Google Scholar
  5. GG81.
    Gabber — Z. Galil: Explicit constructions of linear-sized superconcentrators. J. Comput. Syst. Sci. 22 (1981), 407–420.Google Scholar
  6. GJ76.
    M.R. Garey — D.S. Johnson: Some simplified NP-complete graph problems, Theor. Comp. Science 1 (1976), 237–267.Google Scholar
  7. JAGS85.
    D.S. Johnson — C.R. Aragon — L.A. Mc Geoch — C. Schevon: Optimization by simulated annealing: An experimental evaluation (Part I), Preprint, AT + T Bell Labs, Murray Hill, NY (1985).Google Scholar
  8. KL70.
    B.W. Kernighan — S. Lin: An efficient heuristic procedure for partitioning graphs, Bell Systems Techn. J. 49 (1970), 291–307.Google Scholar
  9. LPS88.
    A. Lubotzky — R. Phillips — P. Sarnak: Ramanujan graphs. Combinatorica 8 (1988), No. 3, 261–277.Google Scholar
  10. MKPR89.
    H. Mühlenbein — O. Krämer — G. Peise — R. Rinn: The Megaframe Hypercluster — A reconfigurable architecture for massively parallel computers, IEEE Conference on Computer Architecture, Jerusalem 1989.Google Scholar
  11. Nic88.
    D.A. Nicole, Esprit Project 1085, Reconfigurable Transputer Processor Architecture, Proc. CONPAR 88, 12–39.Google Scholar
  12. Pet91.
    J. Petersen: Die Theorie der regulären Graphs, Acta Mathematica 15 (1891), 193–220.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Juraj Hromkovič
    • 1
  • Burkhard Monien
    • 1
  1. 1.University of PaderbornPaderbornWest Germany

Personalised recommendations