We give optimally fast $O(\log p)$ time (per processor) algorithms for computing round-optimal broadcast schedules for message-passing parallel computing systems. This affirmatively answers the questions posed in Tr\"aff (2022). The problem is to broadcast $n$ indivisible blocks of data from a given root processor to all other processors in a (subgraph of a) fully connected network of $p$ processors with fully bidirectional, one-ported communication capabilities. In this model, $n-1+\lceil\log_2 p\rceil$ communication rounds are required. Our new algorithms compute for each processor in the network receive and send schedules each of size $\lceil\log_2 p\rceil$ that determine uniquely in $O(1)$ time for each communication round the new block that the processor will receive, and the already received block it has to send. Schedule computations are done independently per processor without communication. The broadcast communication subgraph is the same, easily computable, directed, $\lceil\log_2 p\rceil$-regular circulant graph used in Tr\"aff (2022) and elsewhere. We show how the schedule computations can be done in optimal time and space of $O(\log p)$, improving significantly over previous results of $O(p\log^2 p)$ and $O(\log^3 p)$. The schedule computation and broadcast algorithms are simple to implement, but correctness and complexity are not obvious. All algorithms have been implemented, compared to previous algorithms, and briefly evaluated on a small $36\times 32$ processor-core cluster.

翻译：我们给出了消息传递并行计算系统中计算轮最优广播调度的最优快速$O(\log p)$时间（每处理器）算法。这肯定地回答了Träff（2022）中提出的问题。该问题旨在将$n$个不可分割的数据块从给定的根处理器广播到完全双向、单端口通信能力的$p$个处理器全连接网络（或其子图）中的所有其他处理器。在该模型下，需要$n-1+\lceil\log_2 p\rceil$轮通信。我们的新算法为网络中的每个处理器计算接收和发送调度表，每个调度表大小为$\lceil\log_2 p\rceil$，可以在$O(1)$时间内唯一确定每轮通信中处理器将接收的新块以及需要发送的已接收块。调度计算在每个处理器上独立进行，无需通信。广播通信子图与Träff（2022）及其他文献中使用的相同，为易于计算的有向$\lceil\log_2 p\rceil$正则循环图。我们展示了如何以最优时间复杂度和空间复杂度$O(\log p)$完成调度计算，显著优于此前$O(p\log^2 p)$和$O(\log^3 p)$的结果。调度计算和广播算法实现简单，但其正确性和复杂性并不显而易见。所有算法均已实现，并与先前算法进行了比较，在一个小型$36\times 32$处理器核集群上进行了简要评估。