all reduce

Tensor parallelism divides the weight bytes but not the all-reduce count, so each doubling buys less than the one before. The answer prices that fixed cost, then finds the workload where TP=8 wins anyway.

Two matrix multiplies in a row, and the order of the splits decides whether you communicate once or twice per block. The answer places every split for a 70B layer and prices what the remaining all-reduces send per token.

A ring all-reduce is bandwidth-optimal and takes fourteen synchronized steps on eight GPUs. At 16 KB per call, the steps are the cost. The answer builds the latency-bandwidth model, finds the crossover size, and checks it against the threshold vLLM ships.

Two loops and an index formula, and most attempts get the index wrong by one. The answer writes reduce-scatter and all-gather over a simulated ring, proves the result matches a direct sum when p does not divide the length, and turns the byte count into the reason small messages avoid the ring.