arXiv:2610.31383v1 Announce Type: new Abstract: We study the aggregation task (convergecast) in networks in the case that the values to be aggregated do not fit in a single message. We assume that every node starts with an input value, and there is an aggregation operator that is commutative, associative, and length-preserving, i.e., the result of aggregating two values has the same length as each of the input values. Since values do not fit in a message, they are transferred in a ``train'' of pipelined messages, which must be terminated whenever we want to aggregate values. The goal is to minimize the makespan time of aggregating all values. We present general lower bounds on the problem and optimal algorithms for several prevalent parallel computing architectures, including the hypercube and complete bipartite graphs, under both the splittable and unsplittable flows models. We also present an $O(\log n)$ approximation for general graphs in the unsplittable flows model.