Equation 24 · The Network Is the Computer Again

What does this equation mean?

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

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Inputs and operationslceil log_2 p rceil α + lceil log_2 p rceil n(β + gamma)

Result or conditionT_rd(n,p)

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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T_{\mathrm{rd}}

Symbol T_rd

$T_r$ d is part of the quantity the equation computes from the expression on the right.

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n

Symbol n

n is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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p

Symbol p

p is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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\alpha

Symbol α

the writing.

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\beta

Symbol β

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

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\gamma

Symbol gamma

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

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

Add the term after the plus sign to the term or group before it.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

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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Sources cited in the article section

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