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Equation 30 · Part 4 · What an AI Accelerator Actually Is: Silicon, Packaging, and the Memory It Can Reach

Symbol N

Tallreduce≈2 (p−1) α+2 p−1p⋅Nβlink,T_{\mathrm{allreduce}} \approx 2\,(p-1)\,\alpha + 2\,\frac{p-1}{p}\cdot\frac{N}{\beta_{\mathrm{link}}},
NN

What this part means

the number of bytes.

Its job in the formula

N occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Where the article explains it

For the standard ring formulation of an all-reduce over p devices and N bytes, the operation decomposes into a reduce-scatter followed by an all-gather, each of p - 1 steps in which every device sends N/p bytes.

The passage around this formula

Their cost has a shape worth internalising. For the standard ring formulation of an all-reduce over p devices and N bytes, the operation decomposes into a reduce-scatter followed by an all-gather, each of p - 1 steps in which every device sends N/p bytes. Total time is approximately Tallreduce≈2 (p−1) α+2 p−1p⋅NβlinkT_{\mathrm{allreduce}} \approx 2\,(p-1)\,\alpha + 2\,\frac{p-1}{p}\cdot\frac{N}{\beta_{\mathrm{link}}}. with α\alpha the per-step latency and βlink\beta_{\mathrm{link}} the per-link bandwidth. The bandwidth term…

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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 article section

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