Sparse Relaxed Broadcast Graphs

2026-07-08

AI Summary

This work addresses the problem of constructing sparse n-node graphs in the telephone broadcast model that achieve near-optimal broadcast time—specifically within a factor of (1+ε) of the theoretical lower bound log₂n—while minimizing the number of edges. By augmenting carefully chosen tree structures with a small number of additional edges, the authors strike a balance between communication efficiency and network sparsity. They present a construction requiring only O(n^{1−ε/α}) added edges to attain such near-optimal broadcast time, substantially improving upon the previous upper bound of O(n^{1−ε}), and show this bound is asymptotically tight at interval endpoints. Moreover, they construct graphs with broadcast time ⌈log₂n⌉+1 using at most 2n−4⌈log₂n⌉+O(1) edges, and prove that for infinitely many values of n, achieving this broadcast time necessitates Ω(n) additional edges.

Abstract

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 $ε> 0$, asks for the minimum number of edges of $n$-node graphs with broadcast time close to optimal, i.e., at most $(1+ε)\log_2n$. Let $φ=(1+\sqrt{5})/2$ be the golden ratio, and let $α=1/\log_2φ-1\simeq 0.44$. We show that, for every $n\geq 1$, and for every $ε\in(0,α)$, it suffices to add $O(n^{1-ε/α})$ edges to a well chosen $n$-node tree for designing an $n$-node graph with broadcast time $(1+ε)\log_2n$. This asymptotic bound on the additional number of edges improves the previsouly known bound $O(n^{1-ε})$, 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 $n$, $Ω(n)$ 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 $O(n^{1-ε/α})$ on the additional number of edges for $0<ε<α$ is asymptotically tight at the two extremities of the interval $(0,α]$, as it is $O(n)$ when $ε\to 0$, and $O(1)$ when $ε=α$. 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.

Problem

• broadcasting
• network design
• sparse graphs
• broadcast time
• graph theory

Innovation

• broadcast time
• sparse graphs
• network design
• golden ratio
• telephone model