Concept

Selectivity — where it appears

The fraction of an input a cheap stage proposes to an expensive one, which is the only number deciding whether the cheap stage paid for itself. It appears in no statement of any filtering algorithm and is what every plate about a filter here prints beside the cost of the stage being avoided.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

11010010³10⁴1010010³rows matching the queryblock transfersn/B rows — where the arithmetic says they meetindex, rows scatteredindex, file in key orderread the whole fileB = 64, M = 4,096 (M/B = 64)plus 3 transfers to descend the index

The index that is not worth reading

An index turns a query over 65,536 rows from 1,024 transfers into four. At a thousand matching rows it costs 654 and still wins; at sixteen thousand it costs 1,027 and has lost. Where it turns is decided by the block size — a number the query does not contain, the schema does not mention, and nobody writing either has seen.

applied · Transfer
11010010³10⁴12510rows the query actually matchescost ÷ the better plan's costrho plus the descent, 4.01estimate ÷64estimate ÷8estimate ×8estimate ×64n = 65,536, B = 64, M = 4,096 (M/B = 64), scattered read ×4dotted: the estimate switches at 512 rows

The estimate a plan rests on

A planner chooses between an index and a scan on how many rows it thinks will match, and the number it has is wrong by a factor. Guess sixty-four times too many on a narrow query and the scan it picks costs 13.5 times the index. Guess sixty-four times too few on a wide one and the index costs at most 4.01 times the scan — a ceiling that can be named before any query runs.

applied · Transfer
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
100.0010.010.1fraction of the text proposedlog_4 n = 7.101234,56,7seed length, characters · errors allowed abovepattern 24 · text 20,000 · four symbolsrings: the closed form

The filter that proposes everything

Seed-and-extend saves two thousand times the work at zero errors and costs more than doing nothing at four. Between them the selectivity falls through the floor, and where it falls is set by two numbers that can be computed before the filter is run — one of which does not contain the length of the text at all.

wrong · Distance
rank among the boundaries, sorted by the text before themsorted by the text after0 in the rectanglepattern "ss is un"split after 40 end with the left half0 begin with the rightEnglish-like · 2 copies of 512z = 156 · 0 crossing

The candidates a filter cannot avoid

A phrase index answers a search by intersecting two ranges of boundaries, and it does the intersection by walking the smaller one. On a collection of thirty-two copies that is 4,355 phrase examinations to produce 32 occurrences — 136 examinations each, and rising.

indexes · Grid
trusts the estimateinsured ×2insured ×8always the index11.5235expected regret, logarithmic · label: worst within three standard deviationsrho 14.42.51.11.0rho 215.17.82.22.0rho 413.57.54.04.0rho 1616.016.016.016.0σ = 1.5, median error e^-1, 65,536 rowsexact over the error distribution

What insurance against an estimate costs

A planner that trusts its row estimate expects to pay 1.057 times the better plan and risks 7.76. One that insures itself by halving its estimate before it decides expects 1.057 and risks 4.10 — the insurance is free. At a read ratio of sixteen the same insurance costs six per cent in expectation and makes the worst case worse. Whether a conservative planner is paying a sensible premium depends on two numbers the planner can measure and usually does not — its device's read ratio and the direction its own errors run.

applied · Transfer
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
candidates, filtering4,355candidates, grid33rank and select, grid2,471occurrences32one query · same collection · same answer16,384 characters · 32 occurrencescandidates 132x · operations 1.76x

The operations a candidate count leaves out

The grid examines 33 candidates where the scan examines 4,355 — a factor of 132. Counted in the operations each of them performs, the same query is 2,471 against 4,355, and the factor is 1.8.

wrong · Grid

Named alongside it

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

MeasurementApproximate matchingEdit distanceFilteringSeed-and-extendBlock transferCounting filterCrossoverFalse-positiveHonest limitQ gramQuery plan

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