Equation 24 · Part 4 · The Network Is the Computer Again

Symbol α

T_{\mathrm{rd}}(n,p) = \lceil \log_2 p \rceil \,\alpha + \lceil \log_2 p \rceil \, n\,(\beta + \gamma),

\alpha

What this part means

the writing.

Its job in the formula

α is one of the signed contributions combined to compute the quantity on the left.

Full expressionSymbol αArticle meaning

Where the article explains it

Writing $\alpha$ for latency per message, $\beta$ for transfer time per byte and $\gamma$ for reduction time per byte, the cost of a collective over p participants on n bytes takes the shape T(n,p) = $\alpha$ \,L(p) + $\beta$ \,W(n,p) + $\gamma$ \,C(n,p), where L counts message steps, W counts bytes crossing each link and C counts arithmetic.

The passage around this formula

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

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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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