Error budget — where it appears
Named by 14 essays across 8 fields — each of them below, with the objects they name alongside it.
The branches an error opens
The tree multiplies by 19.6 for the first error, 12.3 for the second, 10.3 for the third and 8.8 for the fourth. A branching factor of four on a sixteen-character pattern would predict sixty-four, and the gap between sixty-four and eight is the intervals emptying.
The search that spends a budget
A backward search narrows one interval per pattern character. Give it a budget of three errors and it narrows 39,943 of them instead, finds every occurrence the whole table finds, and reads not one character of the text — 177,046 index ranks against 60,000 table cells and zero characters examined.
The branch that cannot reach an answer
Seventy-two rank operations over the pattern remove 27,906 of the 39,957 interval extensions a bounded-error index walk performs — 70% of the tree, at a budget of three. The share grows with the budget, which is what a pruning has to do to be worth its cost.
A filter past its design size
A Bloom filter sized for two thousand keys at one per cent answers yes to 15.6% of absent keys at four thousand and 68.1% at eight thousand. Nothing fails and nothing warns. A stack of filters that adds a tighter layer whenever the top one fills holds 2.0% at eight thousand, under a bound it can state in advance — in 2.9 times the bits of one filter sized for eight thousand from the start.
The errors the rest of the pattern needs
Read the pattern left to right in an index of the reversed text and count the points where the interval empties. That count is a lower bound on the errors any alignment of the prefix must contain, it costs 72 rank operations, and it removes 70% of a search tree.
A filter that grows by moving a bit
A table of fingerprints can double in place, moving one stored bit of every fingerprint into its slot number, and so grow as one structure with one lookup where a stack of Bloom filters adds layers. Its false-positive rate is fixed by the fingerprint's length and not by the table, so with nothing reserved it doubles as the keys double — 0.69% at a forecast of 2,000, 5.7% at eight times that. Reserve three bits at the start and it holds 0.66% at eight times, in 294,912 bits, exactly what a table built for sixteen thousand keys would hold and fewer than the stack's 428,938. The reserve is a forecast of growth, and past it the rate climbs again.
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.
A bound that has to be paid for
The pruning removes seventy per cent of a search tree for seventy-two rank operations. It also needs an FM-index of the reversed text — 17,033 bits against the forward index's 17,032 — which doubles the structure whose small size was the entire argument for walking an index.
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.
The pruning that loses an occurrence
Two versions of the same lower-bound pruning, each one character away from correct. Both return only real occurrences, both return fewer of them, and no check that asks whether the answers are right can tell either from the truth.
A schedule nobody writes down
A search scheme is valid when its searches between them cover every way the errors can fall — ten cases at two errors, enumerable in a line. A scheme missing one of them finds seven occurrences of eight, all real, at the right error counts, and reports nothing wrong.
Flat in the budget, and not
One saving is eleven times at every error budget, because it is a property of the alphabet. The other moves between ninety-eight and a hundred and five, because it follows the share of extensions that find nothing. Two savings, two shapes, and neither line crosses the other.
The branches that find nothing
An approximate search over a twenty-symbol alphabet attempts sixteen thousand eight hundred extensions and nine thousand two hundred of them produce an empty interval. That is a full rank walk whose entire result is the discovery that nothing was there.
A looser budget wastes a larger share
More errors permitted means more work, and the fraction of that work which was never going to help rises with it — from thirty-one per cent at no errors to seventy-five at two. The saving is worth most where the search is most expensive.
Named alongside it
The objects these essays reach for when they reach for this one.
Backtracking searchApproximate matchingMeasurementTrade offFM-indexPruningAlphabet sizeEdit distanceLower boundVerificationBackward searchBidirectional index