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

k-wise independence — where it appears

The property that any k keys' hash values are jointly uniform, which is the resource an estimator's variance is bought with and its mean is not. It is the resource an estimator's variance is bought with and its mean is not, so a weaker family leaves the expectation right and the spread unbounded.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Count-Min8,192 bitsCount-Sketch8,192 bitstug-of-war, F22,560 bitsGreenwald–Khanna0 bitsexponential histogram0 bitscash register+strict turnstile±general turnstile± may go negativesliding window+ expiresdeclareddeclared93% under—declareddeclareddeclared—declareddeclareddeclared—declared——————declared40,000 updates · deletion rate 0.5 · every count exact1,758 of 1,895 under

The model a bound was quoted in

Every accuracy figure in this field's first phase was measured under four unstated assumptions. Remove them one at a time and one structure loses its guarantee on 91% of queries, another's error stops falling when it is given more state, and a third has nothing to do at all.

bounds · Bound 1 coefficient13 members2 coefficients169 members3 coefficients2,197 members4 coefficients28,561 members1 key2 keys3 keys4 keys5 keysexact92% gone99% gone100% gone100% goneexactexact92% gone99% gone100% goneexactexactexact92% gone99% goneexactexactexactexact92% goneevery member walked · GF(13) · no tolerance and no seeddegree 1, 2, 3, 4

The independence an estimator spends

Every sketch's analysis begins by assuming a truly random hash, and nobody comes back to that line. Independence has a degree, the degree is enumerable over a small field, and an estimator's mean and its variance spend different amounts of it.

randomness · Moment independent hashestwo values, h₁ + i·h₂2 choices, load 2+60,41859,9942 choices, load 3+2,2832,3672 choices, load 4+123 choices, load 2+46,45646,2433 choices, load 3+138145262,144 keys and buckets, seededbar length is log(1 + count)

Choices that are not independent

The power of two choices is analysed for choices drawn independently, and computing four independent hashes per key costs four hash evaluations. Compute two and take the choices to be h₁, h₁ + h₂, h₁ + 2h₂ and h₁ + 3h₂, and the choices are about as far from independent as they could be. On a million keys the buckets holding two or more come to 147,536 against 147,367 for four independent hashes, and the busiest bucket holds three either way.

randomness · Randomness 3579111311.523510positions per key, krate ÷ the independent rateh₁ + i·h₂h₁ + i·h₂, step oddh₁ + i·h₂ + (i³ − i)/6optimal load m·ln 2 / k; one set of hash functions per schemedashed: the account

Two hash values and the keys they copy

A Bloom filter that makes its k bit positions from two hash values, as h₁ + i·h₂, answers yes to 1.6% of absent keys on a 64-bit filter where k independent hashes answer 0.69%. The penalty is not the one expected. With the step forced odd no key ever repeats a bit, while a quarter of independent keys do. What costs the filter is a query whose start and step reproduce a stored key's whole progression, which happens with probability 4n/m², measured to within a few per cent from 64 bits to 4,096. The penalty fades as the filter grows and returns as the hash count rises — 1.13 times at seven positions on 1,024 bits, 5.35 times at thirteen.

randomness · Randomness

Named alongside it

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

Hash familyIndependence assumptionRandom bitsDerandomisationFalsificationHash functionHonest limitMeasured countUniversal hashingAdversarial inputBloom filterBucket load

All concepts