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

Range reporting — where it appears

Answering which of a stored set of points lie inside a query rectangle, at a cost set by how many there are rather than by how large the rectangle is. It is what a phrase index needs at the centre of its search, where two conditions on the same boundary each pick out an interval of a different ordering.

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

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 phrase table5,148 bitsboundary orders3,744 bitsintersection grid1,696 bitspropagation grid1,544 bitsz = 156 · ⌈log₂ z⌉ = 8 levels16,384 characters · 12,132 bitsgrids 26.7% · 20.8 bits a point

The structure paid for before the first query

The two grids that make a phrase index's search proportional to its answer are 3,240 bits on an 8,892-bit index — twenty-seven per cent of the whole structure, answering nothing on their own, and 38% of them is rank directory rather than payload — the lower-order term of the published bound, measured.

space · Grid 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

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.

structures · Grid 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.

GridOutput-sensitiveBit vectorMeasurementPermutationSelf-indexWavelet treeConstant factorIndex sizeLempel ziv parsePrimary occurrenceRank query

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