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Equation 15 · Part 3 · The Network Is the Computer Again

Symbol p

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 .
pp

What this part means

p is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

p is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

The passage around this formula

…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…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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