Range minimum — where it appears
Named by 18 essays across 8 fields — each of them below, with the objects they name alongside it.
The table that fits inside a block
A block of six parentheses has sixty-four possible shapes and twenty-eight questions can be asked about each, so all 1,792 answers fit in a table of 9,408 bits — computed once, shared by every structure of that block length, and never counted in any of their sizes.
The cost that is the size of the answer
Ten range-minimum queries answer the listing at every point of a sweep where the occurrences run from 30 to 790. They cost 256 to 288 node visits — and the scan they replace costs 30 to 790, so the output-sensitive method loses until about thirty occurrences per document.
What the generated collection was right about
Five strands of conclusions, drawn on collections made by one line with one dial, checked against a corpus nobody made. Most hold. One headline was a property of the generator's alphabet, and one crossing that was guessed at turns out to be met — but only with the structure the strand on range minima built.
Work that falls as the answer grows
Output-sensitive usually means the cost rises with the answer instead of with the input. A descent over a document array costs five operations per document at an answer of seven and two at an answer of thirty-two, because the paths to many leaves share their tops.
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.
Where a crossing moved to
The prediction was that a succinct range minimum would move the document listing's crossing "to a handful". It moves it from 32 occurrences per document to 11 — a factor of three, not an order of magnitude — because a constant-time query is ten lookups rather than one.
Two bits a value, and what undoes them
The parentheses of a range minimum over 65,536 values are 131,072 bits. Everything that makes them answerable is 651,800 more — five times the payload — and one encoding choice nobody quotes accounts for a sixth of it on its own.
The shape a range question is about
A range minimum is a question about a tree, and the tree is determined by the array. Twelve values, eleven parent links, and the answer to every one of the seventy-eight ranges is a lowest common ancestor — with the values themselves no longer needed.
Two thousand documents of two hundred characters
Every collection this field has measured has been a dozen documents of ten thousand characters. A real collection is usually the other shape, and the other shape moves every term in a document index — one of them by a factor of seventy.
The crossing that never arrives
Output-sensitive document listing exists because a pattern can occur four thousand times in eight documents. On a real collection of two thousand short documents it occurs 1.04 times per document, and the whole apparatus buys nothing at all.
The array the walk never reads
Document listing compares a chain entry against the start of a range. The comparison is true exactly when the document has not been reported yet — which the walk already knows, because it just wrote it down.
A document already in the answer
The same walk, with the chain's test replaced by a lookup in the answer so far. It reports the same documents at the same cost, and two exchanged lines make it lose nine of seventeen without failing.
What the chain cost
The chain of previous occurrences is one row number per row — exactly as wide as the suffix array it sits beside, and the largest single part of a document-listing apparatus. It is now absent, and what replaces it is one bit per document.
The tree answers the question
The distinct documents in a range of rows are the distinct symbols of the document array in that range. A wavelet tree enumerates those in one descent, so the range minimum, the chain, the bitmap and the recursion all go at once.
The apparatus that is smaller than its index
Answering "which documents hold this" at a price proportional to the answer used to cost 2.70 times the index it sits beside. Two changes later it costs 0.84, and the largest thing left is an array that says which document each row belongs to.
One copy per document is one occurrence per document
The output-sensitive listing apparatus wins when a pattern occurs far more often than it occurs in documents. A collection of versions was supposed to be that case, and it is the one collection where the two numbers are equal by construction.
The last array in the apparatus
The document-listing apparatus began as three arrays beside a suffix array. Two of them turned out to be machinery for reading the third, and both have gone. What is left is the document array, and it is the only one of the three that was information.
The apparatus, three times smaller again
Eight hundred and thirty-nine thousand bits became two hundred and eighty-five thousand, and now a hundred and twenty thousand. The listing apparatus is fourteen per cent of what it was and holds one array, which is the only part of it that was ever information.
Named alongside it
The objects these essays reach for when they reach for this one.
Document listingIndex sizeOutput-sensitiveDocument arraySuffix arrayDocument collectionMeasurementPrevious occurrence chainConstant factorSpace overheadSuccinctTrade off