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 $\Omega(D)$ 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 $\Omega(k)$ 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 $O(((n+k)/\lambda)\log n)$ rounds in any $n$-node simple graph with edge connectivity $\lambda$. When $k = \Omega(n)$, our algorithm is universally optimal, up to an $O(\log n)$ factor, as its complexity nearly matches an information-theoretic $\Omega(k/\lambda)$ lower bound that applies to all graphs, even when the network topology is known to the algorithm. The setting $k = \Omega(n)$ is particularly interesting because several fundamental problems can be reduced to broadcasting $\Omega(n)$ messages. Our broadcast algorithm finds several applications in distributed computing, enabling $O(1)$-approximation for all distances and $(1+\epsilon)$-approximation for all cut sizes in $\tilde{O}(n/\lambda)$ rounds.

翻译：我们重新审视经典的广播问题：假设网络中有$k$条消息，每条消息由$O(\log{n})$比特组成，这些消息任意分布在网络中。目标是将这些消息广播给所有网络节点。在分布式CONGEST模型中，标准教科书算法在$O(D+k)$轮内解决此问题，其中$D$为图的直径。虽然轮复杂度中的$O(D)$项不可避免（因为任何图中广播问题都需要$\Omega(D)$轮），但$O(k)$项是否在所有图中都是必要的仍不清楚。当最小割大小为1时，仅将消息从割的一侧传输到另一侧就需要$\Omega(k)$轮。然而，若最小割规模更大，则可能设计出更快的算法。这促使我们探索高边连通性网络中的广播问题。本文提出一种简单的随机化分布式算法，可在任意$n$节点且边连通度为$\lambda$的简单图中，用$O(((n+k)/\lambda)\log n)$轮完成$k$-消息广播。当$k = \Omega(n)$时，该算法在$O(\log n)$因子内达到普遍最优，其复杂度几乎匹配信息论下界$\Omega(k/\lambda)$（该下界适用于所有图，即使算法已知网络拓扑）。$k = \Omega(n)$的设定尤为有趣，因为许多基础问题可归结为广播$\Omega(n)$条消息。该广播算法在分布式计算中有多种应用，可在$\tilde{O}(n/\lambda)$轮内实现所有距离的$O(1)$近似以及所有割规模的$(1+\epsilon)$近似。