Algorithms & Data Structures
the bound is a claim, and it is checkable
Concept

Phi function — where it appears

The map sending a text position to the position that precedes it in suffix order. It is linear with slope one between consecutive positions whose rows begin a run, so r anchors and one addition answer it everywhere, and applying it repeatedly turns one known occurrence into all of them.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

text position iphi(i)anchor: a run start25 of thema text that repeats itself · 4 copies of 32r = 26 · pieces 24

A function with r pieces

Computed at every one of eight thousand positions across four texts, a function defined on the whole suffix array agrees exactly with r−1 anchors and one addition. Anchor it at the successor instead of the predecessor — one character of code — and it disagrees at 506 of 800 positions while still returning plausible numbers.

machine · Index patterns of 225%60 selectspatterns of 441%94 selectspatterns of 863%118 selectspatterns of 1682%117 selectsEnglish-like · 8 copies of 512share of steps needing no lookup40 patterns a row

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.

indexes · Index 10,000110sampling, bitssteps per occurrence1 in 21 in 41 in 81 in 161 in 321 in 641 in 128r-indexEnglish-like · 16 copies · r = 233pattern " time "

Every occurrence at the same price

A regular sampling has one dial and it moves two costs together — 1,495 bits at 63 LF steps an occurrence, 131,088 bits at none. A sampling at the run boundaries sits at 10,244 bits and one predecessor query, which is a point the curve reaches only at 69,649.

space · Repeat copy 1copy 8copy 16copy 24copy 32position within the copy32 copies × 2,048 charactersdivergence 0.0005

Sparse on average, dense where it gathers

The r-index keeps one suffix-array value for every run of its transform and prices the set as Elias–Fano: on 32 exact copies of 2,048 characters that is 666 values among 65,537 positions at 9.04 bits a value, as though they were spread evenly. They are not — 663 sit in one copy. Cut into chunks of 32 values, each held by whichever of a bitmap, its own Elias–Fano or a run is cheapest, the same set costs 4.48 bits a value, under the 8.05 that any representation of an arbitrary set of that size must spend on some set. At a divergence of 0.0005 the saving is a fifth, and from one character in five hundred it is nothing but the select directory, which the chunks replace with reading. On the scattered marks of an ordinary index the partition saves only that directory too.

space · Space

Named alongside it

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

R-indexFM-indexIndex sizeLocateMeasurementRun boundary samplingRun-lengthSelf-indexBackward searchBurrows-wheeler transformPredecessor querySampling

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