Efficient construction of broadcast graphs

A. Averbuch, R. Hollander Shabtai, Y. Roditty*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

A broadcast graph is a connected graph, G=(V,E), |V|=n, in which each vertex can complete broadcasting of one message within at most t=⌈logn⌉ time units. A minimum broadcast graph on n vertices is a broadcast graph with the minimum number of edges over all broadcast graphs on n vertices. The cardinality of the edge set of such a graph is denoted by B(n). In this paper we construct a new broadcast graph with B(n)≤(k+1-p)N-(t-k/2 + 2)2k+t-k+2, for n = N == (2k - 1)2t+1-k and B(n) ≤ (k + 1 ? p)n ? (t- k/2 + p + 2)2k + t ? k ? (p ? 2)2 p, for 2t < n < (2k - 1) t+1-k, where t ≥ 7, 2 ≤ k ≤ ⌊ t/2⌋ - 1 for even n and 2 ≤ k ≤ ⌈t/2⌉ - 1 for odd n,d = N ? n, x = ⌊d/2t+1-k⌋ and p = ⌊log2 (x+1)⌋ if x > 0 and p = if x = 0. The new bound is an improvement upon the bounds appeared in Bermond et al. (1995), Bermond et al. (1997), Fertin and Raspaud (2004) and Harutyunyan and Leistman (1999) and the recent bound presented by Harutyunyan and Liestman (2012) for odd values of n.

Original languageEnglish
Pages (from-to)9-14
Number of pages6
JournalDiscrete Applied Mathematics
Volume171
DOIs
StatePublished - 10 Jul 2014

Keywords

  • Broadcasting
  • Minimum broadcast graph

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