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V=2(p−1)⋅Sp≈2Sfor large pV = 2(p-1)\cdot\frac{S}{p} \approx 2S \quad \text{for large } p

Why this formula appears here

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)⋅Sp≈2Sfor large pV = 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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With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article. Read it with the definitions, units, and assumptions supplied by the article.

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V=2(p−1)⋅Sp≈2Sfor large pV = 2(p-1)\cdot\frac{S}{p} \approx 2S \quad \text{for large } p

Equation 6 · Datacenters

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This equation gives an approximation: it relates the quantities while allowing an approximation.

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)⋅Sp≈2Sfor large pV = 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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  • pp: the rather than growing with.
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