Line-broadcasting in complete k -ary trees

Abstract

A line‐broadcasting model in a connected graph G = ( V , E ) , | V | = n , is a model in which one vertex, called the originator of the broadcast, holds a message that has to be transmitted to all vertices of the graph through placement of a series of calls over the graph. In this model, an informed vertex can transmit a message through a path of any length in a single time unit, as long as two transmissions do not use the same edge at the same time. Farley [6] has shown that the process can be completed within at most ⌈ log 2 n ⌉ time units from any originator in a tree (and thus in any connected undirected graph) and that the cumulative cost of broadcasting one message from any vertex, that is, the total number of edges used, is at most ( n − 1 ) ⌈ log 2 n ⌉ . In this article, we present lower and upper bounds for the cumulative cost to broadcast one message in a complete k‐ary tree, k ≥ 2 , from any vertex using the line‐broadcasting model. We prove that if B(u) is the minimum cumulative cost of line‐broadcast in a graph G = ( V , E ) from an originator u ∈ V using the line‐broadcasting model, then ( 2 − o ( 1 ) ) n ≤ B ( u ) ≤ ( 2 + o ( 1 ) ) n , where u is any vertex in a complete k‐ary tree. Furthermore, for certain conditions, B(u) is about ( 2 − o ( 1 ) ) n . © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(2), 139–147 2016