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Broadcast Graphs with Nodes of Limited Memory

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Complex Networks XIII

Abstract

Broadcasting is the process of information dissemination in a network in which a sender, called the originator, wishes to inform all network members as promptly as possible. The broadcast time of a vertex is the minimum time needed to inform all vertices of the network, while the broadcast time of the graph is the maximum broadcast time to broadcast from any originator. A broadcast graph (bg) is a graph with minimum possible broadcast time from any originator. Additionally, a minimum broadcast graph (mbg) is a bg with the minimum possible number of edges. In classical broadcasting, an omniscient who is equipped with adequate memory knows the situation of the whole network as well as the originator in every unit of time. Considering the growth in today’s networks, this is either idealistic in some contexts (such as physical circuits) or at least costly in others (such as telecommunication networks). Consequently, different variations of broadcasting have been suggested in the literature. In this study, we focus on comparing two branches of broadcasting, namely the universal list and messy broadcasting models. To this aim, we propose several general upper bounds for the universal lists by comparing it with the messy broadcasting model. Besides, we propose mbg’s on n vertices for \(n\le 10\) and sparse bg’s for \(11 \le n \le 14\) under universal list model. Afterward, we introduce the first infinite families of bg’s under the universal lists model. Lastly, we prove that hypercubes are mbg under the universal lists.

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Notes

  1. 1.

    Or broadcast algorithm.

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A Appendix

A Appendix

The universal lists achieving broadcast time of \(\lceil \log n \rceil \) under the fully-adaptive model for the mbg’s and bg’s on n vertices presented in Fig. 1:

  • \(\sigma _7=\{1:<6,2>,2:<3,1>,3:<4,2,7>,4:<3,5>,5:<6,4>,6:<1,5,7>,7:<3,6>\}\),

  • \(\sigma _9=\{1:<6,2>,2:<3,1>,3:<4,2,9>,4:<3,5>,5:<6,4>,6:<1,5,7>,7:<6,8>,8:<7,9>,9:<3,8>\}\),

  • \(\sigma _{10}=\{1:<2,8>,2:<1,3,6>,3:<2,4>,4:<5,10,3>,5:<4,6>,6:<2,5,7>,7:<8,6>,8:<9,1,7>,9:<8,10>,10:<9,4>\}\).

  • \(\sigma _{11}=\{1:<7,8,2,10>, 2:<6,3,1>, 3:<4,2>, 4:<3,5,9>, 5:<11,4,6>, 6:<5,2,7>, 7:<6,1>, 8:<9,11,1>, 9:<8,4>, 10:<11,1>, 11: <5,10,8>\}\),

  • \(\sigma _{12}=\{1:<2,12,7>,2:<8,3,1>,3:<2,9,4>,4:<3,5>,5:<11,6, 4>,6:<12,7,5>,7:<6,1,8>,8:<2,9,7>,9:<8,3,10>,10:<11, 9>,11:<12,5,10>,12:<6,1,11>\}\),

  • \(\sigma _{13}=\{1:<2, 6, 8>, 2:<1, 3, 9>, 3:<2, 4, 7, 10>, 4:<3, 5, 11>, 5:<4, 6, 12>, 6:<5, 1, 7, 13>, 7:<9, 6, 11, 3, 13>, 8:<13, 9, 1>, 9:<8, 10, 2, 7>, 10:<9, 11, 3>, 11:<10, 12, 4, 7>, 12:<11, 13, 5>, 13: <12, 8, 6, 7>\}\),

  • \(\sigma _{14}=\{1:<6,2,8>,2:<3,1,9>,3:<4,2,7,10>,4:<3,5,11>,5:<6,4,12>,6:<1,5,7,13>,7:<3,6,14>,8:<13,9,1>,9:<10,8,2>,10:<11,9,14,3>,11:<10,12,4>,12:<13,11,5>,13:<8,12,14,6>,14:<10,13,7>\}\).

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Gholami, S., Harutyunyan, H.A. (2022). Broadcast Graphs with Nodes of Limited Memory. In: Pacheco, D., Teixeira, A.S., Barbosa, H., Menezes, R., Mangioni, G. (eds) Complex Networks XIII. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-031-17658-6_3

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