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

Quantile summary — where it appears

A structure answering any quantile of a stream from a few stored values, with a promise about the rank of the answer rather than about its size. Its promise is about the rank of the answer rather than about the value, and the two are unrelated: a small rank error can be a large error in the number returned.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

quantilepromisedone summarymerge of 8q = 0.5100.01418q = 0.920.0183q = 0.992.005overq = 0.9990.214overrank error, in items out of 20,000ε = 0.01 · high-biased · 8 shards, round2 of 4 quantiles over the single summary's promise

The promise that does not survive the tree

A high-biased quantile summary at ε = 1% answers the 99th percentile of twenty thousand values to within two ranks. Merge thirty-two of them and the answer is seventeen ranks out — inside the merged bound, eight times outside the one anybody was quoted.

wrong · Rank 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 0231436845165232shards mergedkeys whose count movedthree orders:folded in one at a timecombined pairwise, in a treefolded in, last shard firstMisra-Gries, k = 32 · hashedworst gap 298 arrivals

What a fold charges per level

Thirty-two counter tables folded in a chain come out wrong by 323 where the same thirty-two combined pairwise are wrong by 148, and the quantile summaries prefer the chain by exactly as much in the other direction. What is being charged in each case is the depth of the fold, and the two families are charged on opposite ones.

bounds · Merge 1,000248163264high-biased, α = 0.56low-biased, α = 0.56none-biased, α = 0.74tuples keptshards mergedlog-normal, σ = 1.2 — a latency distribution · 20,000 valuesα 0.56 / 0.56 / 0.74 · worst residual 6.5%

The cheap tail and the expensive merge

A summary whose tolerance tightens towards the tail keeps seven times the tuples of a plain one on a single pass, and after merging sixty-four shards it keeps three and a half times as many. The error function that buys a useful tail promise is also the one that pays most for never having the values in one place.

space · Rank answered 4,1900%25%50%75%100%11.794,190value, logarithmicfraction of the stream at or belowrank ±2%value 24.9–4,190answered 1553.5% out20,000 values · Pareto, α = 1.2 — a heavy tail · Greenwald–Khanna, ε = 0.02rank 0.10% · value 1553.5%

A promise about the rank is not a promise about the value

A quantile summary asked for the 99th percentile of a log-normal stream returns the largest value it ever saw — 2,169 against a true 318, six times too high — and its rank error is 1.00% against a promised 2%. The guarantee held. It was never about the number.

wrong · Rank ε = 0.0515 → 29 (1.93×)ε = 0.0238 → 73 (1.92×)ε = 0.0177 → 136 (1.77×)ε = 0.005152 → 270 (1.78×)ε = 0.002397 → 674 (1.70×)reportedoccupied at the peak20,000 arrivals · lognormalpeak = resident + period, to 14%

The tuples a summary does not report

A Greenwald–Khanna summary at ε = 0.01 answers `tuples` with seventy-seven. Watched through the run it holds a hundred and thirty-six. The gap is the compression period, it is 1.70 to 1.93 times across every tolerance measured, and it is the number a deployment has to allocate.

space · Space 1,00010,000248163264ε = 0.02, α = 0.58ε = 0.01, α = 0.56ε = 0.005, α = 0.54tuples keptshards mergedlog-normal, σ = 1.2 — a latency distribution · 20,000 valuesα 0.58 / 0.56 / 0.54 · worst residual 2.0%

The tuples a merge does not give back

A merge of thirty-two quantile summaries keeps seven times the tuples of one summary over the same values, and sixty-four keeps ten and a half. Fitted across the sweep the count goes as the shard number to the power 0.56, which answers what it converges to — it does not.

streaming · Rank 10010³10⁴10⁵10⁶⌊1/2ε⌋ = 50151025501002005001,000tuples, and tuples examinedcompression period, in updatestuples examinedpeak tuplesresident tuplesworst rank errorε = 0.01 · 20,000 arrivalspeak 10× · work 72× · answer 1.21×

The period that is not a promise

Greenwald–Khanna's ε appears twice — once as the rank tolerance the structure promises, and once as ⌊1/2ε⌋, the number of updates between compressions. Unhook the second from the first and sweep it across a thousand-fold range. The tuples held move by 10%, the worst rank error by 21%, the peak by ten times and the housekeeping by seventy.

streaming · Rank folded in one at a time2,616 tuples17 ranks outcombined pairwise, in a tree3,637 tuples17 ranks outfolded in, last shard first2,615 tuples17 ranks outtuples kept, and worst rank error against a promise of 10032 shards · ε = 0.01 · high-biased · round1.39× the space, 0 ranks of answer

The shape that moves the bill

Thirty-two quantile summaries combined pairwise keep 3,637 tuples and the same thirty-two folded in one at a time keep 2,616, for answers that differ by nothing at all. The counter tables measured for the same thing do the opposite — their order moves the answer and leaves the space alone.

structures · Rank counter tablesworst error, ratio to bestquantile summariestuples kept, ratio to bestchain2.18× (323)1.10× (2,807)tree1.00× (148)1.26× (3,211)smallest-first1.24× (183)1.28× (3,278)largest-first2.72× (403)1.00× (2,556)32 shards · hashed · k = 32each column against its own best shape

The shape one structure will not fold

Folding thirty-two shards largest-pair-first keeps 2,556 quantile tuples against a balanced tree's 3,211 — a fifth of the space saved. The same fold on the counter tables beside them leaves 403 counts of error against the tree's 148. A deployment holding both cannot fold once and be right twice.

structures · Merge 1 — no differencethe whole array reviewed1.03×only this batch reviewed1.22×a block window, which does alias1.48×20,000 arrivals · period 50 · cyclethe instrument reads 1.48 where an alias is known to be

The sampler that cannot alias

A block window's boundary fires on an arrival count, and a stream whose burst repeats every sixty-four arrivals is reported as perfectly even by a block of five hundred and twelve. A quantile summary compresses on an update count, which is the same arrangement. Swept against three periodic value processes and their shuffles, it does not alias — and the reason is one line of arithmetic rather than a lucky sweep.

wrong · Window

Named alongside it

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

Greenwald–KhannaGuaranteeRank errorState bitsTrade offMergeable summaryShardMerge treeSpace overheadMeasurementCompression scheduleEstimator

All concepts