Concept

Data movement — where it appears

Bytes copied rather than compared, a cost that separates two sorts of identical comparison count and decides which is faster. It separates two sorts of identical comparison count and decides which finishes first, which is the second count arriving in a third guise.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

block transfersIn order, 0 to n−11,0241.0× a scan · 64.0 elements per transferEvery B-th element (B = 64)65,53664.0× a scan · 1.0 elements per transferUniformly at random61,40760.0× a scan · 1.1 elements per transfera scan of this array is 1,024 transfersB = 64, M = 4,096 (M/B = 64)64× between the cheapest order and the dearest

One access, eight kilobytes

Every count on this site charges one for an array access. A machine charges for a block. The same 65,536 accesses cost 1,024 transfers in one order and 65,536 in another, with nothing about the algorithm's work changed — a factor of 64, which is exactly the number of elements in a block, and which no counter here could see until now.

applied · Transfer
comparisonsswapsInsertion sort63,071 / 0Selection sort130,816 / 504Bubble sort129,688 / 62,563Merge sort3,964 / 0Heapsort7,653 / 4,170Quicksort5,049 / 2,380n = 512, random inputcounted in the same run

One run, four counts, four answers

The question “how many operations” has no answer until the operation is named. Selection sort makes more comparisons than any other algorithm here and fewer writes than almost all of them; bubble sort matches its comparisons and does 124 times the swapping. The ranking depends entirely on which count is chosen, and the choice needs justifying.

counting · Count
elements of block written per key insertedB-tree, in place49.3Log-structured, T = 23.0 · 16× less than the treeLog-structured, T = 42.0 · 25× less than the treeLog-structured, T = 81.0 · 49× less than the treeLog-structured, T = 161.0 · 49× less than the treeB = 64, M = 4,096 (M/B = 64)25× between the two structures at T = 4

The writes nobody counted

Sixteen thousand keys inserted into a B-tree write 49.3 elements' worth of blocks for every key stored. The same keys into a log-structured store write 2.0. Every operation counter reports the two as the same work — the same insertions, the same comparisons, the same number of updates — and the factor of 24 decides which structure a storage engine is built from.

applied · Transfer
1010010³10⁴B — elements to the blockblock transfersthey cross near B = 4one element at a timesorted by destination, 2passesM = 4,096, B as drawnthe lower bound is the smaller of the two, and it is proved rather than measured

Permuting is the harder problem here

Rearranging 65,536 elements into a stated order costs 63,601 transfers one at a time and 4,096 by sorting them into place. In the model every other field on this site uses, the first is the cheap method and beats the second by a factor of eight. The two models disagree about which problem is easy, and they disagree by about the same factor in opposite directions.

applied · Transfer
4 bytes32 bytes128 bytes512 bytesrecord:QuicksortMerge sortShellsortHeapsortInsertion sortSelection sortBubble sortQuicksortShellsortMerge sortHeapsortSelection sortInsertion sortBubble sortQuicksortShellsortMerge sortSelection sortHeapsortInsertion sortBubble sortSelection sortQuicksortShellsortMerge sortHeapsortInsertion sortBubble sort1234567n = 512, random input, key 8 bytesa read or a write moves the record; a comparison touches the key

The exchange rate nobody wrote down

Three earlier essays have said in passing that the ranking would change if the elements were large records. None of them computed it. Computed, selection sort goes from second-worst of seven at four bytes a record to best of seven at five hundred and twelve — and the crossover against each rival is a division that takes one line.

counting · Count

Named alongside it

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

Cost modelBlock transferExternal-memory modelTrade offComparison countLocalityMemory hierarchyOperation countRankingScanB-treeCache

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