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

Compound walk — where it appears

The operation returning a symbol's rank and the count of every smaller symbol from one descent of a wavelet tree, at the price of the rank alone. The second number is what a bidirectional extension needs and had been computed by summing a rank per alphabet symbol.

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

025507510000.50011.502errors permittedfactor against the published loop64.5%55.8%74.7%the walk: 11xthe descentthe label is the shareof dead extensions6 patterns · sigma 2198x to 105x

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.

bounds · Distance a loop over the alphabet1,478,400the compound walk, inside the loop134,40011xone descent per node18,91678x6 patterns · 1 error · sigma 2155.8% of the extensions were dead

Three savings on one structure

A factor of eleven on an extension, a factor of seven on a branching search, and a sixth of the bits. Applied to one bidirectional index they do not give a factor of seventy-seven, and the reason is that two of the three are the same saving.

indexes · Distance factor against the published loopthe compound walk11xthe enumeration78xthe two, multiplied860xnothing measures thisan index with both78xthe shortfall is 11x, and the walk's factor is 11x6 patterns · 1 error · sigma 21the larger of the two, not the product

Two factors that do not multiply

Eleven times and seventy-eight times against the same baseline, so an index with both should be eight hundred and sixty. It is seventy-eight, and the shortfall is eleven — the first factor, exactly, because the second operation already contains it.

wrong · Distance a loop over the alphabet107.9the compound walk10.0the cheapest hereone descent per step19.016 exact patterns of 12ranks per step · sigma 21

The saving that is a loss

An operation that is seventy-eight times cheaper on a branching search costs twice as much on an exact one. It reports every symbol present in order to hand back the one that was asked for, and a search that knows its character needs none of the rest.

practice · Distance

Named alongside it

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

Bidirectional indexInterval symbolsBacktracking searchCompositionRankWavelet treeAlphabet sizeBackward searchCheckDead branchError budgetIndex size

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