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Broadcast Graphs Using New Dimensional Broadcast Schemes for Knödel Graphs

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Algorithms and Discrete Applied Mathematics (CALDAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10156))

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Abstract

Broadcasting is an information disseminating process of distributing a message from an originator vertex v of graph \(G=(V,E)\) to all of its vertices. The broadcast time of vertex v is the minimum possible time required to broadcast from v in graph G. A graph G on n vertices is called broadcast graph if broadcasting from any originator in G can be accomplished in \(\lceil \log n\rceil \) time. A broadcast graph with the minimum number of edges is called minimum broadcast graph. The number of edges in a minimum broadcast graph on n vertices is denoted by B(n). Finding the values of B(n) is very difficult. A large number of papers present techniques to construct broadcast graphs and to obtain upper bounds on B(n). In this paper, we first find new dimensional broadcast schemes for Knödel graphs, and then use them to give a general upper bound on B(n) for almost all odd n.

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References

  1. Averbuch, A., Shabtai, R.H., Roditty, Y.: Efficient construction of broadcast graphs. Discret. Appl. Math. 171, 9–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barsky, G., Grigoryan, H., Harutyunyan, H.A.: Tight lower bounds on broadcast function for n = 24 and 25. Discret. Appl. Math. 175, 109–114 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bermond, J.-C., Fraigniaud, P., Peters, J.G.: Antepenultimate broadcasting. Networks 26, 125–137 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bermond, J.-C., Harutyunyan, H.A., Liestman, A.L., Perennes, S.: A note on the dimensionality of modified Knödel graphs. Int. J. Found. Comput. Sci. 8(02), 109–116 (1997)

    Article  MATH  Google Scholar 

  5. Bermond, J.-C., Hell, P., Liestman, A.L., Peters, J.G.: Sparse broadcast graphs. Discret. Appl. Math. 36, 97–130 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chau, S.C., Liestman, A.L.: Constructing minimal broadcast networks. J. Comb. Inf. Syst. Sci. 10, 110–122 (1985)

    MathSciNet  MATH  Google Scholar 

  7. Dinneen, M.J., Fellows, M.R., Faber, V.: Algebraic constructions of efficient broadcast networks. In: Mattson, H.F., Mora, T., Rao, T.R.N. (eds.) AAECC 1991. LNCS, vol. 539, pp. 152–158. Springer, Heidelberg (1991). doi:10.1007/3-540-54522-0_104

    Chapter  Google Scholar 

  8. Dinneen, M.J., Ventura, J.A., Wilson, M.C., Zakeri, G.: Compound constructions of broadcast networks. Discret. Appl. Math. 93, 205–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Farley, A.M.: Minimal broadcast networks. Networks 9, 313–332 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Farley, A.M., Hedetniemi, S., Mitchell, S., Proskurowski, A.: Minimum broadcast graphs. Discret. Math. 25, 189–193 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fertin, G., Raspaud, A.: A survey on Knödel graphs. Discret. Appl. Math. 137, 173–195 (2004)

    Article  MATH  Google Scholar 

  12. Gargano, L., Vaccaro, U.: On the construction of minimal broadcast networks. Networks 19, 673–689 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grigni, M., Peleg, D.: Tight bounds on mimimum broadcast networks. SIAM J. Discret. Math. 4, 207–222 (1991)

    Article  MATH  Google Scholar 

  14. Grigoryan, H., Harutyunyan, H.A.: New lower bounds on broadcast function. In: Gu, Q., Hell, P., Yang, B. (eds.) AAIM 2014. LNCS, vol. 8546, pp. 174–184. Springer, Heidelberg (2014). doi:10.1007/978-3-319-07956-1_16

    Google Scholar 

  15. Harutyunyan, H.A.: An efficient vertex addition method for broadcast networks. Internet Math. 5(3), 197–211 (2009)

    MathSciNet  Google Scholar 

  16. Harutyunyan, H.A., Li, Z.: A new construction of broadcast graphs. In: Govindarajan, S., Maheshwari, A. (eds.) CALDAM 2016. LNCS, vol. 9602, pp. 201–211. Springer, Heidelberg (2016). doi:10.1007/978-3-319-29221-2_17

    Chapter  Google Scholar 

  17. Harutyunyan, H.A., Liestman, A.L.: More broadcast graphs. Discret. Appl. Math. 98, 81–102 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harutyunyan, H.A., Liestman, A.L.: Improved upper and lower bounds for k-broadcasting. Networks 37, 94–101 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harutyunyan, H.A., Liestman, A.L.: Upper bounds on the broadcast function using minimum dominating sets. Discret. Math. 312, 2992–2996 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Harutyunyan, H.A., Liestman, A.L., Peters, J.G., Richards, D.: Broadcasting and gossiping. In: Handbook of Graph Theorey, pp. 1477–1494. Chapman and Hall (2013)

    Google Scholar 

  21. Hedetniemi, S.M., Hedetniemi, S.T., Liestman, A.L.: A survey of gossiping and broadcasting in communication networks. Networks 18, 319–349 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khachatrian, L.H., Harutounian, H.S.: Construction of new classes of minimal broadcast networks. In: Conference on Coding Theory, Dilijan, Armenia, pp. 69–77 (1990)

    Google Scholar 

  23. Knödel, W.: New gossips and telephones. Discret. Math. 13, 95 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Labahn, R.: A minimum broadcast graph on 63 vertices. Discret. Appl. Math. 53, 247–250 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maheo, M., Saclé, J.-F.: Some minimum broadcast graphs. Discret. Appl. Math. 53, 275–285 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mitchell, S., Hedetniemi, S.: A census of minimum broadcast graphs. J. Comb. Inf. Syst. Sci. 5, 141–151 (1980)

    MathSciNet  MATH  Google Scholar 

  27. Park, J.-H., Chwa, K.-Y.: Recursive circulant: a new topology for multicomputer networks. In: International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN), pp. 73–80. IEEE (1994)

    Google Scholar 

  28. Saclé, J.-F.: Lower bounds for the size in four families of minimum broadcast graphs. Discret. Math. 150, 359–369 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shao, B.: On K-broadcasting in graphs. Ph.D. thesis, Concordia University (2006)

    Google Scholar 

  30. Ventura, J.A., Weng, X.: A new method for constructing minimal broadcast networks. Networks 23, 481–497 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Weng, M.X., Ventura, J.A.: A doubling procedure for constructing minimal broadcast networks. Telecommun. Syst. 3, 259–293 (1994)

    Article  Google Scholar 

  32. Xiao, J., Wang, X.: A research on minimum broadcast graphs. Chin. J. Comput. 11, 99–105 (1988)

    Google Scholar 

  33. Zhou, J., Zhang, K.: A minimum broadcast graph on 26 vertices. Appl. Math. Lett. 14, 1023–1026 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Concordia University, Montreal, Québec, H3G 1M8, Canada

    Hovhannes A. Harutyunyan & Zhiyuan Li

Authors
  1. Hovhannes A. Harutyunyan
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  2. Zhiyuan Li
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Correspondence to Zhiyuan Li .

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Editors and Affiliations

  1. University of Lethbridge, Lethbridge, Alberta, Canada

    Daya Gaur

  2. Indian Institute of Technology Madras, Chennai, India

    N.S. Narayanaswamy

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Harutyunyan, H.A., Li, Z. (2017). Broadcast Graphs Using New Dimensional Broadcast Schemes for Knödel Graphs. In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_18

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  • DOI: https://doi.org/10.1007/978-3-319-53007-9_18

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  • Print ISBN: 978-3-319-53006-2

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