Broadcasting in one dimension |
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Affiliation: | School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6 |
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Abstract: | We consider the broadcasting problem for one-dimensional grid graphs with a given neighborhood template. There are two different models that have been considered-shouting (a node informs all of its neighbors in one step) and whispering (a node informs a single neighbor in one step). Let σ(t) (respectively ω(t)) denote the maximum number of nodes that can be reached in t steps by shouting (respectively whispering) broadcast from a single source.We obtain detailed information about the benefits of shouting over whispering. We prove for the one-dimensional case a conjecture by Stout that ω(t) eventually becomes a polynomial. In particular, we show that there exist constants i and t0 such that ω(t)=σ(t)−i for all t ≥ t0. When the broadcast only goes in one direction (i.e., when all elements of the template are positive), we also determine that i=d −1 and t0≤3d for a neighborhood template with the furthest neighbor at distance d. |
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