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

Elias fano — where it appears

A representation of a sorted set of positions storing each one's low bits packed and its high bits as a bit vector of bucket boundaries. It costs about m(2 + log(n/m)) bits for m elements in a universe of n, and answers select in one operation and rank in one plus a bounded walk.

Named by 10 essays across 7 fields — each of them below, with the objects they name alongside it.

1,00010,00010⁴10⁵characters in the collectionbitsFM-index, plainFM-index, compressedrun-length indexsample one in 64 · divergence 0 · r = 22411,900 bits at 32 copies

The index that stores the runs

A compressed self-index over thirty-two copies of a text is 30,557 bits, because its size follows an entropy that cannot see a copy. An index that stores the transform as its runs is 11,900 — and at a single copy it is the larger of the two, which is what makes the comparison a claim about repetition rather than about size.

indexes · Repeat 01e+42e+43e+44e+450100one sampled position in every …bitsplain, one bit a rowcompressed blocksElias-Fano16,384 rowscrosses at one in 4

Twenty bits apart

Two representations of one sparse set, six thousand seven hundred and forty-five bits against six thousand seven hundred and sixty-five. One exploits sparsity and the other exploits runs, and on this set at this density they price identically.

machine · Space 05e+31e+41.5e+40246bits kept in the low partbits⌊log₂(n/m)⌋ = 2low partshigh vector2,049 marks in 16,385 positions1 bits between the two roundings

The flat bottom of a shallow curve

The low width is chosen as the floor of log of the universe over the count. Rounding it up instead costs one bit on five thousand, because the total is m·w plus n over two to the w and the minimum is where those two are equal.

bounds · Space 02040255075100125one sampled position in every …of the index, per centplain marks: 18.5%Elias-Fano: 3.8%16,384 characters14.9% off at one in 32

What the locating apparatus becomes

The two parts that answer "where" are half an index at a dense sampling and a fifth at a sparse one, and the fifth does not fall further. Represent the marks properly and it keeps falling, to under four per cent.

indexes · Space plain bit vector19,475compressed blocks4,55723.4%Elias–Fano3,90520.1%positions, priced as Elias–Fano3,59118.4%16,385 rows · one in 32 markedElias-Fano at 20.1%

A position split in two

Write each sorted position as a high part and a low part. Store the low parts packed and the high parts as a bit vector in which the k-th one sits at position (p >> w) + k. A select on that vector and a low read recover any position.

structures · Space both halves locate250,584100.0%the reverse half counts only226,50490.4%and its marks are Elias-Fano210,93484.2%the saving spent on sampling241,47596.4%16,384 characters · one in 3284.2% smaller, or the same size and faster

The ladder, and the rung that spends

Two hundred and fifty thousand bits, then two hundred and twenty-six, then two hundred and eleven. The fourth rung takes the whole saving and buys a four-times denser sampling with it, landing at ninety-six per cent of where it started and locating several times faster.

space · Distance 01e+42e+43e+44e+450100one sampled position in every …bitsplain, one bit a rowcompressed blocksElias-Fano16,384 rowscrosses at one in 4

Where the sparse representation loses

At every row marked, Elias–Fano costs twice the plain vector. The crossing is at one row in four, which is a sampling rate a real index uses — so the choice between the two is a choice, not an improvement.

wrong · Space bits heldthe model3,591the structure3,905one rank, in operationsbinary search10 stepsselect and walk1 select + 0.7016,385 rows · one in 32 marked14x on the operation

A price with no structure under it

A class in this collection has charged Elias–Fano's price for several strands and stores an array of positions searched by binary search. The accounting is right about space to a bit per thousand and wrong about one operation by a factor of eight.

wrong · Space 0102030100200one sampled position in every …of both halves saved, per centplain marks: 13.5%Elias-Fano: 1.8%16,384 characters17.6x against 2.3x

The floor was the marks

A saving reported as about a sixth of a bidirectional index, falling to an eighth and levelling off. Represent one array properly and it falls to a fiftieth instead — most of what was being dropped was a badly encoded bit vector.

practice · Space 020406080255075100125one sampled position in every …of the plain index, per centsize: 84.7%67.5 steps16,384 characters · 8-character patternssize falls, the walk lengthens

Bits and steps on one frame

The size falls from ninety-nine per cent to eighty-six as the sampling thins, and the walk to a sampled position rises from two and a half steps to sixty-four. Neither line is the answer; the answer is a point on the pair.

structures · Distance

Named alongside it

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

Index sizeSample marksLocatingSpace accountingSparse setBidirectional indexBit vectorRankSelf-indexSuffix array samplingCheckCrossing point

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