# Asymptotic Bounds You May Not Have Learned In School

in mathematics •  2 months ago

Computer Science education teaches "Big O" notation for describing complexity upper bounds, and Big-Theta and Big-Omega get introduced as well. Mathematicians and complexity theorists sometimes use "little o" as a stronger description of asymptotic behavior. But here are some bounds you may not have learned.

## Big-D notation

`f(x)` is `D(g(x))` if you plot the two on Desmos, and it looks like `f(x) < g(x)`.

Example: `n^r` is `D( log(n) )` for small values of `r`, even though `log(n)` is `O(n^r)`for any positive `r`.

This notation is surprisingly popular with many students, perhaps due to trouble distinguishing a handwritten 'O' from a handwritten 'D'. Unfortunately, the broader mathematical community is resistant to this innovation in asymptotic analysis.

## little-w asymptotic bound

`f(x)` is `w(g(x))` if you hand-wave hard enough that the two are equivalent.

Example: `2^(n/2)` is `w(2^n)`, because all running times that use an exponent are the same.

This notation frequently gets used in the context of NP-complete problems where it asserted that the best known running time is `w(2^n)`. (Upper bounds on 3SAT are actually as low as `O(1.321^n)` for a randomized algorithm.)

## curly-O notation

The curly-O notation is the same as big-O, but reserved for those who want to show off their typesetting or calligraphy skills.

Example:

## QWT-boundedness

A decision function that could be computed in time `f(n)` on a quantum computer because it "explores all universes simultaneously" is `QWT(f(n))`

(Editor's note: that's not how quantum computers work.)

## Œ notation

A randomized procedure's run time is `Œ( g(x) )` if the bound on its expected run time is `O(g(x))`, or if the function is described in Old English.

Example: lookup in a hash table takes `Œ(1)` time. It does not take `O(1)` time as sloppy writing sometimes indicates.

Example: the notorious Beowulf reduction demonstrates the complexity class `M_onsters` are all `Œ(Geats)`. "Hwæt. We Gardena in geardagum, þeodcyninga, þrym gefrunon, hu ða æþelingas ellen fremedon...."

## Shrug relation

`f(x) ¯\_(ツ)_/¯ g(x)` if either `f(x)` is asymptotically less than `g(x)`, or `g(x)` is asymptotically less than `f(x)`, but we're not sure which, or the proof is too complicated to explain.

Example: The Slow-growing hierarchy `g_α(n)` eventually catches up with the Fast-growing hierarchy `f_α(n)`, so `f_α(n) ¯\_(ツ)_/¯ g_α(n)` because who understands infinite ordinals anyway.

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The curly-O notation is the same as big-O, but reserved for those who want to show off their typesetting or calligraphy skills.

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