Sparse Relaxed Broadcast Graphs

Broadcasting in graphs refers to the information dissemination problem in which a source node has an atomic piece of information to be distributed to all the nodes of a graph. In the standard telephone model, broadcasting proceeds as a sequence of synchronous rounds, where, at each round, every informed node can transfer the information to at most one of its neighbors. The broadcast time of a graph $G$ is the maximum, taken over every node $v\in V(G)$, of the minimum number of rounds required for broadcasting from $v$ in $G$. We study the network design problem that, for every $\epsilon> 0$, asks for the minimum number of edges of $n$-node graphs with broadcast time close to optimal, i.e., at most $(1+\epsilon)\log_2n$. Let $\phi=(1+\sqrt{5})/2$ be the golden ratio, and let $\alpha=1/\log_2\phi-1\simeq 0.44$. We show that, for every $n\geq 1$, and for every $\epsilon\in(0,\alpha)$, it suffices to add $O(n^{1-\epsilon/\alpha})$ edges to a well chosen $n$-node tree for designing an $n$-node graph with broadcast time $(1+\epsilon)\log_2n$. This asymptotic bound on the additional number of edges improves the previsouly known bound $O(n^{1-\epsilon})$, and has implications to the design of graphs with minimum broadcast cost, defined as number of edges times broadcast time. Moreover, we show that, for infinitely many values of $n$, $\Omega(n)$ edges must be added to some tree for designing an $n$-node graph with broadcast time $\lceil\log_2 n\rceil+1$. Therefore, our bound $O(n^{1-\epsilon/\alpha})$ on the additional number of edges for $0<\epsilon<\alpha$ is asymptotically tight at the two extremities of the interval $(0,\alpha]$, as it is $O(n)$ when $\epsilon\to 0$, and $O(1)$ when $\epsilon=\alpha$. Finally, we show that, for every $n$, there exists an $n$-node graph with broadcast time $\lceil\log_2 n\rceil+1$ and at most $2n-4\lceil\log_2n\rceil+O(1)$ edges.