multiplication

The passage around this formula

The ring algorithm is the one worth understanding mechanically, because it is bandwidth-optimal and because its cost model exposes exactly why topology matters. In a ring all-reduce across p workers, each worker is logically placed on a ring; a gradient buffer of size S is split into p chunks, and each worker simultaneously sends one chunk to its ring-neighbor while receiving a different chunk from its other neighbor, reducing (summing) as chunks arrive. A complete all-reduce takes 2(p-1) such steps, each moving S/p bytes [ 5 ] . The total data volume any one worker sends over the whole operation is: $V = 2(p-1)\cdot\frac{S}{p} \approx 2S \quad \text{for large } p$. which is the reason ring all-reduce is called bandwidth-optimal: in the…

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