Equation 9 · The Network Is the Computer Again
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
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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Symbol T
T is part of the quantity the equation computes from the expression on the right.
Symbol n
n is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol p
p is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol L
L is one of the signed contributions combined to compute the quantity on the left.
Symbol β
β is one of the signed contributions combined to compute the quantity on the left.
Symbol W
W is one of the signed contributions combined to compute the quantity on the left.
Symbol gamma
gamma is one of the signed contributions combined to compute the quantity on the left.
Symbol C
C is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
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What the article says around this equation
The classical accounting for a collective operation separates three terms: a per-message latency, a per-byte transfer cost, and a per-byte computation cost for reductions [ 2 ] . Writing for latency per message, for transfer time per byte and for reduction time per byte, the cost of a collective over p participants on n bytes takes the shape . where L counts message steps, W counts bytes crossing each link and C counts arithmetic. The whole art of collective algorithm design is choosing which of these terms to pay.
Sources cited in the surrounding passage
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