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Published equation contexts

Trd(n,p)=⌈log⁡2p⌉ α+⌈log⁡2p⌉ n (β+γ)T_{\mathrm{rd}}(n,p) = \lceil \log_2 p \rceil \,\alpha + \lceil \log_2 p \rceil \, n\,(\beta + \gamma)

Why this formula appears here

But look at the latency term: 2(p-1)α\alpha grows linearly in the number of participants. A recursive-doubling or tree scheme instead completes in ⌈\lceil log⁡2\log_2 p ⌉\rceil rounds: Trd(n,p)=⌈log⁡2p⌉ α+⌈log⁡2p⌉ n (β+γ)T_{\mathrm{rd}}(n,p) = \lceil \log_2 p \rceil \,\alpha + \lceil \log_2 p \rceil \, n\,(\beta + \gamma). paying a logarithmic latency term but sending the full message in every round, so its bandwidth cost grows with log⁡\log p rather than staying flat.

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TrdT_{\mathrm{rd}}

Symbol T_rd

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

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Published contexts (1)

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Trd(n,p)=⌈log⁡2p⌉ α+⌈log⁡2p⌉ n (β+γ),T_{\mathrm{rd}}(n,p) = \lceil \log_2 p \rceil \,\alpha + \lceil \log_2 p \rceil \, n\,(\beta + \gamma),

Equation 24 · 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.

But look at the latency term: 2(p-1)α\alpha grows linearly in the number of participants. A recursive-doubling or tree scheme instead completes in ⌈\lceil log⁡2\log_2 p ⌉\rceil rounds: Trd(n,p)=⌈log⁡2p⌉ α+⌈log⁡2p⌉ n (β+γ)T_{\mathrm{rd}}(n,p) = \lceil \log_2 p \rceil \,\alpha + \lceil \log_2 p \rceil \, n\,(\beta + \gamma). paying a logarithmic latency term but sending the full message in every round, so its bandwidth cost grows with log⁡\log p rather than staying flat.

Meanings in this article

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