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Tring(n,p)=2(p−1) α+2(p−1)p n β+p−1p n γT_{\mathrm{ring}}(n,p) = 2(p-1)\,\alpha + \frac{2(p-1)}{p}\,n\,\beta + \frac{p-1}{p}\,n\,\gamma

Why this formula appears here

Consider the two canonical all-reduce algorithms. A ring arranges participants in a cycle and performs a reduce-scatter followed by an all-gather, each in p-1 steps carrying n/p bytes: Tring(n,p)=2(p−1) α+2(p−1)p n β+p−1p n γT_{\mathrm{ring}}(n,p) = 2(p-1)\,\alpha + \frac{2(p-1)}{p}\,n\,\beta + \frac{p-1}{p}\,n\,\gamma . The bandwidth term converges to 2nβ\beta as p grows — it stops depending on the number of participants — which is why the ring is the bandwidth-optimal choice for large messages. NVIDIA’s own performance documentation encodes exactly this factor, defining the bus bandwidth of an all-reduce by applying a correction of 2(p-1)/p to the naive size-over-time figure, on the reasoning that an all-reduce requires 2(p-1) data transfers across p links; all-gather, reduce-scatter and all-to-all get a…

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TringT_{\mathrm{ring}}

Symbol T_ring

TrT_ring is part of the quantity the equation computes from the expression on the right.

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Tring(n,p)=2(p−1) α+2(p−1)p n β+p−1p n γ.T_{\mathrm{ring}}(n,p) = 2(p-1)\,\alpha + \frac{2(p-1)}{p}\,n\,\beta + \frac{p-1}{p}\,n\,\gamma .

Equation 15 · AI Hardware & Semiconductors

The Network Is the Computer Again

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Consider the two canonical all-reduce algorithms. A ring arranges participants in a cycle and performs a reduce-scatter followed by an all-gather, each in p-1 steps carrying n/p bytes: Tring(n,p)=2(p−1) α+2(p−1)p n β+p−1p n γT_{\mathrm{ring}}(n,p) = 2(p-1)\,\alpha + \frac{2(p-1)}{p}\,n\,\beta + \frac{p-1}{p}\,n\,\gamma . The bandwidth term converges to 2nβ\beta as p grows — it stops depending on the number of participants — which is why the ring is the bandwidth-optimal choice for large messages. NVIDIA’s own performance documentation encodes exactly this factor, defining the bus bandwidth of an all-reduce by applying a correction of 2(p-1)/p to the naive size-over-time figure, on the reasoning that an all-reduce requires 2(p-1) data transfers across p links; all-gather, reduce-scatter and all-to-all get a…

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