← All parts of this equation

Equation 15 · Part 12 · The Network Is the Computer Again

Numerator: 2(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 .
2(p−1)2(p-1)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

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

The passage around this formula

…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 factor of (p-1)/p , while broadcast and reduce get a factor of one because everything must pass through a single root [ 3 ] . The point of the correction is that bus bandwidth should stay roughly constant as ranks…

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.