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

Tail behaviour — where it appears

What a cost distribution does at its far end, as against what it does on average. It is the quantity a system with a deadline is limited by, it is invisible to any benchmark that reports a mean, and two methods with identical averages can differ in it by a growing factor.

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

share of nodespredictedlevel 149.68% · 50.00%level 225.29% · 25.00%level 312.52% · 12.50%level 46.24% · 6.25%level 53.08% · 3.13%level 61.64% · 1.56%level 70.79% · 0.78%level 80.34% · 0.39%20,000 nodes, p = 0.5, seed 5150worst departure 0.32 points

The height is a distribution, and the coin is a parameter

A skip list over 2,048 keys is described as being about log₂ n levels tall. Across two hundred builds of exactly those keys its height ranged from 9 to 19. The number in the description is the mean of something, and choosing the coin is choosing which something.

randomness · Randomness 024681,0244,09616,38465,536262,144keys, into as many bucketskeys in the busiest bucketone hashtwo hashes, take the emptierthree hasheslog n / log log nthe average load is 1 at every pointa lookup examines every choice, so two hashes is two probes

The second choice

Two hundred and sixty thousand keys into as many buckets. Under one hash the busiest bucket holds eight; under two, with each key going to whichever of its two is emptier, it holds four. The mean is exactly one in both. Nothing is rearranged afterwards, no key is ever moved, and the whole of the improvement is in a decision taken once, at the moment the key arrives.

randomness · Randomness 2 halves, ties go left2 choices, ties at randomone choiceload 2 or more14,61015,03717,363load 3 or more2675785,250load 4 or morenone11,236load 5 or morenonenone223load 6 or morenonenone36load 7 or morenonenone2load 8 or morenonenone165,536 keys and buckets, seededbar length is log(1 + count)

The tie that breaks left

Two choices per key, the emptier bucket wins, and when the two are equally full a coin decides. Replace the coin with a rule — split the table into halves and always send a tie to the left one — and on a million keys the buckets holding three or more fall from 9,316 to 4,694, and the busiest bucket drops from four to three. The hashing, the probes and the keys are unchanged, and the rule spends no randomness at all.

randomness · Randomness 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

Named alongside it

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

Bucket loadConcentrationHash functionMaximum loadMeasured countTwo choicesHash familyLoad balancingRandom bitsRandomised data structureTrade offDerandomisation

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