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)


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 \(k>1\).


Bipartite Graph Binary Sequence Complete Bipartite Graph Read Operation Arbitrary Graph 


  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

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