Fast Broadcast in Highly Connected Networks

TL;DR

This paper introduces a simple randomized CONGEST algorithm that solves $k$-broadcast in $O\big(\frac{n+k}{\lambda}\log n\big)$ rounds on any $n$-node graph with edge connectivity $\lambda$, and shows that for $k=\Theta(n)$ this bound is universally optimal up to a $\tilde{O}(\log n)$ factor by matching a $\Omega(k/\lambda)$ information-theoretic lower bound. The key technical tool is a low-diameter tree-packing decomposition: random edge sampling yields $\Omega(\lambda/\log n)$ edge-disjoint spanning subgraphs with diameter $O\big( (n\log n)/\delta \big)$, enabling parallelized, low-congestion dissemination. Leveraging this decomposition, the authors obtain fast distributed algorithms for approximate All-Pairs Shortest Paths (APSP) and for approximating all cuts via spectral sparsifiers, achieving sublinear rounds in graphs with high edge connectivity. The results provide near-optimal, universally-bounded performance for fundamental dissemination tasks in highly connected networks and offer practical reductions to compute distances and cuts more efficiently in distributed settings.

Abstract

We revisit the classic broadcast problem, wherein we have $k$ messages, each composed of $O(\log{n})$ bits, distributed arbitrarily across a network. The objective is to broadcast these messages to all nodes in the network. In the distributed CONGEST model, a textbook algorithm solves this problem in $O(D+k)$ rounds, where $D$ is the diameter of the graph. While the $O(D)$ term in the round complexity is unavoidable$\unicode{x2014}$given that $Ω(D)$ rounds are necessary to solve broadcast in any graph$\unicode{x2014}$it remains unclear whether the $O(k)$ term is needed in all graphs. In cases where the minimum cut size is one, simply transmitting messages from one side of the cut to the other would require $Ω(k)$ rounds. However, if the size of the minimum cut is larger, it may be possible to develop faster algorithms. This motivates the exploration of the broadcast problem in networks with high edge connectivity. In this work, we present a simple randomized distributed algorithm for performing $k$-message broadcast in $O(((n+k)/λ)\log n)$ rounds in any $n$-node simple graph with edge connectivity $λ$. When $k = Ω(n)$, our algorithm is universally optimal, up to an $O(\log n)$ factor, as its complexity nearly matches an information-theoretic $Ω(k/λ)$ lower bound that applies to all graphs, even when the network topology is known to the algorithm. The setting $k = Ω(n)$ is particularly interesting because several fundamental problems can be reduced to broadcasting $Ω(n)$ messages. Our broadcast algorithm finds several applications in distributed computing, enabling $O(1)$-approximation for all distances and $(1+ε)$-approximation for all cut sizes in $\tilde{O}(n/λ)$ rounds.

Theorems & Definitions (48)

• Definition 1: $k$-broadcast
• Theorem 1
• Theorem 2
• proof
• Lemma 1: Basic broadcast algorithm peleg2000distributed
• Lemma 2: Breadth-first search peleg2000distributed
• Lemma 3
• proof
• Lemma 4
• proof