Algorithms & Data Structures
the bound is a claim, and it is checkable
Concept

Scale — where it appears

How large a measurement’s object was when it was taken. It belongs beside any per-unit quantity whose own axis appears inside it — phrases per character carry a logarithm of the length, so a rate measured at one size mispredicts another by tens of per cent.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

123456123456cost to open a gapcost to extend a gapintentionexecution571 settingsinte---ntion---execution3 settingsinte-ntion-execution1 setting-intentionexec-ution1 settingintention against execution, affine costseach colour is one optimal alignment

The parameter plane has few answers

Sweep the cost of opening a gap against the cost of extending one over five hundred and seventy-six settings, and the optimal alignment of intention against execution takes four values — one of them at 571 of the settings. Under a linear model the plane divides into three wedges through the origin, because doubling every cost changes nothing and only the ratio is a parameter. Tuning an aligner is choosing a region, and most of the plane is one.

tables · Cost ① prose144,617 chars12 docs · median 12,335① code106,283 chars8 docs · median 13,150① revisions75,658 chars10 docs · median 7,774② essays1,004,822 chars82 docs · median 12,335② captions549,210 chars2,214 docs · median 230② history174,282 chars14 docs · median 12,963① the first freeze · ② the second6 parts

A million characters of the same thing

Every measurement this collection has published about real text was taken on twenty-four thousand characters, because the phrase count was quadratic. It is linear now, so here is the same corpus at forty times the size — and what forty times does to its own numbers.

text · Document 10⁴10⁵10⁶characters in the collectionshuffled ÷ real, phraseswhat the order is wortha million characters of prose · phrases per characterx1.37 over the sweep

The half of a fall that is the logarithm

Phrases per character on a real collection of essays fall by a factor of 2.34 as it grows. A shuffle of the same characters falls by 1.70. Nearly three quarters of the movement is arithmetic, and no definition of the measure says so.

wrong · Document ① prose12,33512① code13,1508① revisions7,77410② essays12,33582② captions2302,214② history12,96314documentscharacters in a document① the first freeze · ② the secondmedian marked

Two thousand documents of two hundred characters

Every collection this field has measured has been a dozen documents of ten thousand characters. A real collection is usually the other shape, and the other shape moves every term in a document index — one of them by a factor of seventy.

indexes · Document 11010⁴cap on the depthphrasesno capcap 165,536 characters · 16 copies26x at cap 1

What a quadratic construction was setting

A depth cap of one costs 90% of an 8,192-character text in phrases, and 86% of a 65,536-character one. The ladder's conclusions hold at thirty-two times the size — and the number the ladder could not reach, the deepest chain a real collection produces, turns out to be fourteen.

wrong · Parse

Named alongside it

The objects these essays reach for when they reach for this one.

AlphabetCorpusRepetitionControlDocument collectionEntropyIndex sizeLogarithmPhrase countRun countAffine gapAlignment

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