Rank query — where it appears
Named by 10 essays across 6 fields — each of them below, with the objects they name alongside it.
A search that runs backwards
Twenty-three occurrences of a six-character pattern in sixteen thousand characters, found in twelve rank queries and zero character comparisons. Not few comparisons — none. The algorithm never asks whether two symbols are equal, and it knows how many matches there are before it has located one.
The index that is smaller than the text
The Burrows–Wheeler transform is a permutation, so it changes no symbol frequency and a plain index over it is the same size whether the text has deep structure or none — 6.29 bits a character against 6.16, on texts whose third-order entropies differ fourfold. What the transform changed was the runs, and a structure that charges one bit per bit cannot see a run.
Rank is the only thing it does
Constant time and o(n) extra space — a phrase true of a rank directory costing 163% overhead and reading three words, and equally true of one costing 3% and reading eighteen. Both numbers are decided by two integers somebody typed into a header, and the phrase names neither.
The text that does not have to be kept
The index reproduces its text character for character, in 1,024 mapping steps and zero reads of anything. That is the whole justification for weighing it against the text rather than beside it — and the price is a dial that moves the structure by 3.3 times and the cost of locating one match by 72.
The occurrence carried through the search
Backward search returns how many and not where, and every index on this site pays for the second question separately. Carrying one occurrence along with the interval costs a lookup on 75% of the steps for a two-character pattern and on 18% of them for a sixteen-character one, and it is what makes a run-boundary sampling usable at all.
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 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 list of documents is not a list of occurrences
A pattern occurring 790 times in eight documents has an answer of size eight. Reading every occurrence to find out costs 790 array reads; the question a collection has that a text does not is the one its index does not answer.
A rectangle over a permutation
Two orderings of one set of boundaries are two permutations, so a phrase index's intersection is a rectangle over a permutation grid — the one point set a wavelet tree stores exactly, at one bit per point per level and no coordinates at all.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
MeasurementSelf-indexIndex sizeBackward searchBit vectorWavelet treeBurrows-wheeler transformHonest limitTrade offFM-indexOutput-sensitivePattern matching