Advertisement

Sequences of Radius k for Complete Bipartite Graphs

  • Michał Dębski
  • Zbigniew Lonc
  • Paweł Rzążewski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9941)

Abstract

Let G be a graph. A k-radius sequence for G is a sequence of vertices of G such that for every edge uv of G vertices u and v appear at least once within distance k in the sequence. The length of a shortest k-radius sequence for G is denoted by \(f_k(G)\).

Such sequences appear in a problem related to computing values of some 2-argument functions. Suppose we have a set V of large objects, stored in an external database, and our cache can accommodate at most \(k+1\) objects from V at one time. If we are given a set E of pairs of objects for which we want to compute the value of some 2-argument function, and assume that our cache is managed in FIFO manner, then \(f_k(G)\) (where \(G=(V,E)\)) is the minimum number of times we need to copy an object from the database to the cache.

We give an asymptotically tight estimation on \(f_k(G)\) for complete bipartite graphs. We show that for every \(\epsilon >0\) we have \(f_k(K_{m,n})\le (1+\epsilon )d_k\frac{mn}{k}\), provided that both m and n are sufficiently large – where \(d_k\) depends only on k. This upper bound asymptotically coincides with the lower bound \(f_k(G)\ge d_k\frac{e(G)}{k}\), valid for all bipartite graphs.

We also show that determining \(f_k(G)\) for an arbitrary graph G is NP-hard for every constant \(k>1\).

Keywords

Bipartite Graph Binary Sequence Complete Bipartite Graph Read Operation Arbitrary Graph 
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.

References

  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method, 3rd edn. Wiley, Hoboken (2008)CrossRefMATHGoogle Scholar
  2. 2.
    Bertossi, A.A.: The edge Hamiltonian path problem is NP-complete. Inf. Proc. Lett. 13, 157–159 (1981)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blackburn, S.R.: The existence of \(k\)-radius sequences. J. Combin. Theor. Ser. A 119, 212–217 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blackburn, S.R., McKee, J.F.: Constructing \(k\)-radius sequences. Math. Comput. 81, 2439–2459 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chee, Y.M., Ling, S., Tan, Y., Zhang, X.: Universal cycles for minimum coverings of pairs by triples, with applications to 2-radius sequences. Math. Comput. 81, 585–603 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chinn, P., Chvátalová, J., Dewdney, A., Gibbs, N.: The bandwidth problem for graphs and matrices - a survey. J. Graph Theor. 6, 223–254 (1982)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dębski, M., Lonc, Z.: Sequences of large radius. Eur. J. Comb. 41, 197–204 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Frankl, P., Rödl, V.: Near perfect coverings in graphs and hypergraphs. Eur. J. Comb. 6, 317–326 (1985)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. Freeman, New York (1979)MATHGoogle Scholar
  10. 10.
    Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM J. Appl. Math. 34, 477–495 (1978)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jaromczyk, J.W., Lonc, Z.: Sequences of radius k: how to fetch many huge objects into small memory for pairwise computations. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 594–605. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Jaromczyk, J., Lonc, Z., Truszczyński, M.: Constructions of asymptotically shortest \(k\)-radius sequences. J. Combin. Theor. Ser. A 119, 731–746 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Newman, A.: Max-cut. Encycl. Algorithms 1, 489–492 (2008)CrossRefGoogle Scholar
  14. 14.
    Poljak, S., Tuza, Z.: Maximum cuts and large bipartite subgraphs. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 20, 181–244 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  • Michał Dębski
    • 1
  • Zbigniew Lonc
    • 2
  • Paweł Rzążewski
    • 2
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

Personalised recommendations