Concept

Q gram — where it appears

A substring of exactly q characters, counted with multiplicity over a pattern or a window. One error destroys at most q of them, which is what lets a count of shared q-grams decide whether a window is worth verifying.

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

k = 0k = 1k = 2k = 3k = 4k = 5k = 6q = 223211917151311q = 3221916131074q = 4211713951-3q = 520151050-5-10q = 6191371-5-11-17q = 81791-7-15-23-31a shaded cell is a threshold of zero or less: every window proposedt = m + 1 − q(k+1) · m = 2411 collapsed cells

The q-grams an error cannot destroy

A pattern of twenty-four characters holds twenty-one four-grams. Two errors can destroy at most eight of them, so any occurrence with two errors still shares thirteen — and a filter that keeps only the windows sharing thirteen proposes 104 of 3,977 and computes 16,744 table cells instead of 96,000.

text · Distance
0.010.11errors allowed, kshare of windows proposed01234q = 3q = 4q = 5a shaded dot is a collapsed thresholdm = 24 · n = 4,000 · 6 planted1 collapsed rows

The threshold that reaches zero

At q = 5 and four errors on a twenty-four-character pattern the filter demands zero shared q-grams, proposes all 2,977 windows, and computes 986,266 table cells where filling the whole table would have cost 96,000. The failure is arithmetic and is knowable before a character is read.

wrong · Distance
the whole table60,00060,000r · 0rkcounting filter24,84315,369r · 0rkseed filter17,7452,415r · 1,377rkindex walk00r · 177,046rkn = 3,000 · m = 20 · q = 4cells drawn · r = characters read, rk = index rankscells computed6 occurrences

Three savings in three currencies

The same six occurrences, found four ways. The table computes 60,000 cells and reads 60,000 characters. The counting filter computes 6,251 cells and reads 13,022 characters. The seed filter computes 3,325 and reads 550. The index walk computes none, reads none, and performs 18,645 ranks.

wrong · Distance

Named alongside it

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

Approximate matchingCounting filterEdit distanceFilteringMeasurementSeed-and-extendSelectivityVerificationFalse-positiveUnit of costAlphabet sizeBacktracking search

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