Concept

Backtracking — where it appears

Walking an index with an error budget, branching over the alphabet wherever an error is spent. The tree it explores multiplies by roughly ten per error on the alphabets measured here, which is what every schedule and every lower bound in this field exists to cut.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

characters examinedNaive scan520,25663.51 per text characterKnuth–Morris–Pratt16,3211.99 per text characterBoyer–Moore–Horspool8,1290.99 per text characterRabin–Karp00.00 per text characterone unit = one character comparisonone matcher read no characters at all

The shift the pattern already knows

On a text of 8,192 characters the naive scan examines 520,256 of them and Knuth–Morris–Pratt examines 16,321 — a factor of 32, and 16,321 is 99.6% of the 2n that no input can push it past. On ordinary random text the same two algorithms examine 20,862 and 20,833. Both measurements are of the same pair of algorithms and only one of them is the reason anybody uses the second.

text · Symbol
right to left8302.74xleft to right8772.89xone per piece4471.48xthe scheme3031.00xinterval extensionsall four found the same 6 positions15 characters · k = 22.74x apart

The search that starts in the middle

The same pattern, the same six occurrences, the same index — and 830 interval extensions, or 303, according to which end the search begins at. A pattern cut into three pieces has an error-free one, and only a search with two ends can start there.

text · Distance
1010³10⁴10⁵10⁶characters of textstepsbacktrackingThompson's NFAopen points: capped at4,000,000(a|a)*b12,158x at 20 characters

The folklore is about a matcher

Four million steps against three hundred and twenty-nine, on a twenty-character expression matched against twenty characters. One of the two machines doubles with every character of the input and the other does not, and only one of them is what a regular expression is.

wrong · Automaton

Named alongside it

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

AlphabetAlphabet sizeAmortised analysisApproximate matchingAutomatonBidirectional indexCharacter comparisonConstant factorError budgetExponential timeFailure functionFilter

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