The tuple count of a merged summary, fitted

The tuple count of a merged summary, fitted

Tuples kept by the merge of m summaries of 20,000 values, on log axes, so a power law is a straight line and the exponent is the slope. ε = 0.02: exponent 0.58, multiplier 283 against a single summary's 277; ε = 0.01: exponent 0.56, multiplier 527 against a single summary's 512; ε = 0.005: exponent 0.54, multiplier 942 against a single summary's 958. The multiplier is the single-pass tuple count to within 3%, so the shape is tuples(m) ≈ tuples(1)·m^α. Halving ε moves the multiplier and leaves the exponent where it was, so ε buys resolution and the shard count buys nothing but tuples.

Drawn at 700 × 396, wide on the page. Everything above is what quantile-tuples returns with no arguments; the caption is the generator's own, computed from the numbers in the drawing rather than written beside it.

4 essays call quantile-tuples. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about — which is what optcheck and figfill exist to catch.