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

Partial order — where it appears

What a set of comparisons establishes: some pairs ordered, others not yet, and everything implied by transitivity. The arrangements still consistent with it are its linear extensions, and counting them is how much a sort has left to find out.

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

1416642561k2k4k8k16k33k66k11010010³10⁴10⁵10⁶keys handed backcomparisons no method can go underthe losers, and the outputs' ownorderevery element not handed back losesthe answer is one of n!/(n−k)!sequences65,536 keyseach is a claim about every possible method

Two floors that can be added

Handing back the ten smallest of 65,536 keys has two floors under it and neither is close where they cross — the larger of the two is 63,821 comparisons at k = 4,000 and the best method makes 125,401. They can be added, because a comparison that eliminates a key the caller never sees can never be a comparison that orders two the caller does see. Charged together the floor rises 62%, and the tournament goes from 1.97 times it to 1.21.

counting · Count 0%25%50%75%Merge sort0.0% · 1.03× the floorQuicksort, random pivot0.0% · 1.16× the floorQuicksort, first-element pivot0.0% · 1.18× the floorMerge sort with an insertion cutoff0.0% · 1.31× the floorInsertion sort0.0% · 9.83× the floorShellsort9.8% · 1.31× the floorQuicksort, median-of-three15.4% · 1.18× the floorHeapsort26.2% · 1.45× the floorBubble sort49.2% · 9.83× the floorSelection sort59.9% · 7.77× the floorshare of comparisons whose answer was already implied256 elements, random orderlight: repeats nothing

The questions a sort asks twice

Selection sort makes 32,640 comparisons on 256 elements and 19,561 of them have answers it already holds. Remove every one and it still makes 7.8 times the information floor, because a question can be new and nearly worthless: insertion sort repeats nothing at all and removes 0.72 of a bit per comparison where merge sort removes 0.96. And bubble sort, less its repeats, makes exactly insertion sort's comparisons — at every size.

counting · Count 32641282565121010010³10⁴elements sortedword operations a comparison must cost to break evenSelection sort, pairsSelection sort, closureBubble sort, pairsBubble sort, closureHeapsort, pairsHeapsort, closureQuicksort, median-of-three, pairsQuicksort, median-of-three, closuresolid: pairs asked · dashed: everything impliedrandom input · counted exactly

The price of remembering an answer

Give selection sort a table of the pairs it has already compared and it makes 16,805 comparisons on 256 elements instead of 32,640. The table pays for itself once a comparison costs more than 3.1 word operations, and that price stays near three at every size measured. Give it the full closure of everything its answers imply and it makes 13,079. The closure pays only past 425 operations a comparison at that size, and past 1,659 at twice the size. The cheap memory is worth having on long keys. The complete one is worth having almost nowhere.

counting · Count 11010010³10⁴10⁵10⁶k, the number handed backcomparisonstournament, measuredfloor that names the outputsbest earlier floorn = 65,536, one shuffled inputthe floors are worst-case, the count is one input

The comparisons that name the answer

Returning the 4,000 smallest of 65,536 keys in order needs 61,536 comparisons to eliminate the rest and 42,100 to order the ones returned. That was the floor, 103,636, and a tournament made 125,388. What the floor never charged is saying which 4,000 come back. Charge that, and the floor is 125,341. On the same input the tournament is 47 comparisons above it, and for every k up to a hundred it is exactly on it.

counting · Count

Named alongside it

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

Comparison countHonest limitInformation floorLower boundAdversary argumentExhaustive searchFalsificationOutput-sensitiveRedundancySelectionSelection sortTransitivity

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