SP-Allreduce: Symmetric Tree Schedules for Tori and Beyond

Abstract

Allreduce is a fundamental communication primitive in large-scale distributed training, and torus interconnects remain widely deployed in high-performance systems. We introduce SP-allreduce, a new allreduce algorithm for $d$-dimensional tori that exploits the symmetry of the topology and routes only along shortest paths. We call an allreduce algorithm \emph{fully bandwidth-optimal} if it incurs no bandwidth overhead, no multihop-induced traffic inflation, and achieves perfect per-round load balance. SP-allreduce is based on a family of carefully symmetrized shortest-path trees that guarantees full bandwidth optimality in the $α$--$β$ model. We prove that it uses the minimum possible number of rounds among all fully bandwidth-optimal allreduce algorithms on $d$-dimensional tori, and therefore the minimum running time in this class. In this sense, SP-allreduce refines Bucket allreduce: it preserves Bucket's key strengths while reducing the number of rounds by a factor of two. We further extend our approach to broader symmetric topologies and to communication trees of prescribed depths. Experimentally, the resulting algorithms improve over the best compared state-of-the-art algorithms for tori by up to $1.19\times$ on square tori and $1.65\times$ on rectangular tori across the tested configurations. On HyperX and spine-leaf networks, we demonstrate gains of up to $14.82\times$ and $1.72\times$, respectively, over the best evaluated baselines.