Source-Oblivious Broadcast

TL;DR

This work shows that restricting broadcast protocols to a source-oblivious, fully-adaptive encoding does not diminish the ability to design $n$-node broadcast graphs with optimal $\lceil \log_2 n\rceil$ rounds, while enabling near-optimal sparsity. It provides two constructive results: (i) for every $n$, there exist $n$-node graphs with $b_{fa}(G)=\lceil \log_2 n\rceil$ and (ii) there exist such graphs with $|E|=O(n\cdot L(n))$, where $L(n)$ is the number of leading 1s in the binary of $n-1$. The constructions leverage (a) a hypercube-based scheme to achieve fast broadcast with compact per-node lists and (b) binomial-tree-based assemblies to minimize edges while preserving optimal broadcast time. The results bridge the gap between fast dissemination and space-efficient protocol encoding, and raise open questions about adaptive and non-adaptive source-oblivious models and the precise edge-minimum for optimal broadcast graphs.

Abstract

This paper revisits the study of (minimum) broadcast graphs, i.e., graphs enabling fast information dissemination from every source node to all the other nodes (and having minimum number of edges for this property). This study is performed in the framework of compact distributed data structures, that is, when the broadcast protocols are bounded to be encoded at each node as an ordered list of neighbors specifying, upon reception of a message, in which order this message must be passed to these neighbors. We show that this constraint does not limit the power of broadcast protocols, as far as the design of (minimum) broadcast graphs is concerned. Specifically, we show that, for every~$n$, there are $n$-node graphs for which it is possible to design protocols encoded by lists yet enabling broadcast in $\lceil\log_2n\rceil$ rounds from every source, which is optimal even for general (i.e., non space-constrained) broadcast protocols. Moreover, we show that, for every~$n$, there exist such graphs with the additional property that they are asymptotically as sparse as the sparsest graphs for which $\lceil\log_2n\rceil$-round broadcast protocols exist, up to a constant multiplicative factor. Concretely, these graphs have $O(n\cdot L(n))$ edges, where $L(n)$ is the number of leading~1s in the binary representation of $n-1$, and general minimum broadcast graphs are known to have $Ω(n\cdot L(n))$ edges.

Figures (4)

• Figure 1: Graph in the proof of Proposition \ref{['prop:lafamille']}, for $k=2$.
• Figure 2: Construction in the proof of Theorem \ref{['theo:BG']}. For every $i$, $\alpha_i,\beta_i,\gamma_i,\delta_i\in\{0,1\}$.
• Figure 3: Broadcast from source node $u$
• Figure 4: Broadcast from source node $v_{m-i}$

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Corollary 1
• Proposition 2
• proof
• proof : Proof of Theorem \ref{['theo:BG']}
• proof : Proof of Theorem \ref{['theo:MBG']}