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

Deterministic algorithm — where it appears

An algorithm making no random choices, so its guarantee holds on every input rather than on most of them and no adversary has anything to aim at. It is what derandomisation buys, and the price is usually a constant factor plus the loss of any defence against an input chosen to hurt it.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

hash rowh1h2h3h4h1 → cell 30: 10,433h2 → cell 29: 10,624h3 → cell 19: 10,575h4 → cell 4: 10,895key 0 occurred 9,579 times · the minimum of the four is 10,433 · over by 854the additive bound at this width is e/w × N = 5,0974×32 counters · 4,096 bits · Zipf s = 1.1exact would take 162,288 bits

A count that is never under

The Count-Min sketch holds four rows of counters and answers how often a key occurred. Its error is one-sided with no probability attached — the estimate is never below the truth on any stream — and the probabilistic half of its guarantee is only about how far above.

streaming · Sketch keytruthfoldedtreereversed16,3626,8666,8696,86923,0743,5783,5813,58131,9632,4672,4702,47041,3731,9021,9071,95451,0981,6021,6051,60569351,5121,4711,46377691,2801,2851,30686561,1601,1631,16395661,1001,1011,095105101,0511,0361,029shaded rows are keys whose count depends on the order aloneSpace-Saving, k = 32 · 8 shards · hashed37 keys move with the order

The order nobody fixed

The same eight summaries, combined pairwise in a tree or folded in one at a time, produce tables that differ on forty keys — the largest by 964 arrivals. Nothing in a deployment fixes which shape is used, and both answers are inside the guarantee.

structures · Merge answered 19.90%25%50%75%100%0.36420.52,170value, logarithmicfraction of the stream at or belowrank ±2%value 19.3–21.9answered 2.9% out20,000 values · log-normal, σ = 1.2 — a latency distribution · Greenwald–Khanna, ε = 0.02rank 1.02% · value 2.9%

The error that is on the rank

A summary of 77 tuples answers eight quantiles of a stream of 20,000 values, and every answer is guaranteed to sit within 0.9% of the stream from where it was asked for. The guarantee is deterministic, it holds on every distribution, and it is not about the numbers it returns.

streaming · Rank key 05,416short by 4,163key 1462short by 4,154key 31short by 2,099key 71short by 943key 19401short by 3counter held (bar) against true count (tick)12 counters · 768 bits · no randomnessbound N/(k+1) = 4,615

The items that survive k counters

Misra-Gries keeps k counters, decrements all of them on a miss, and never returns a count above the truth — with no hashing, no randomness and no failure probability. At equal state it is more accurate than the randomised sketch on the question both are usually asked, at every size measured.

structures · Sketch HyperLogLog1024/1024 identicalCount-Min256/256 identicalbottom-k128/128 identicalMisra-Gries20/36 identicalfraction of the state that merged to the identical value→ 1,974→ 10,291→ 2,322→ 8,766two streams of 30,000 · 2,007 distinct keys in the union3 of 4 merge exactly

The summaries that add

Two sketches built over two streams and merged are, for three of the four structures here, byte for byte the summary the concatenated stream would have produced. For the fourth the guarantee survives and the state does not, and calling both properties mergeability hides the difference that matters.

structures · Cardinality 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

Named alongside it

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

EstimatorGuaranteeSketchState bitsCount-Min sketchHeavy hitterMergeable summaryMisra–GriesAdditive errorHash functionIdempotenceOne pass

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