← All parts of this equation

Equation 15 · Part 13 · The Network Is the Computer Again

Numerator: p-1

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 .
p−1p-1

What this part means

The complete quantity above the fraction bar.

Its job in the formula

p-1 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

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,…

Read this part in the article →

Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.