Access pattern — where it appears
Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.
The count is not the time
An operation count is exact, machine-independent, and not a running time. The gap between them is mostly memory, and it is large enough to reorder the rankings. This site carries a second count — modelled cache misses from the same runs — and asserts that the two disagree, because if they agreed the second one would carry no information.
The cliff where the data stops fitting
Below the cache's capacity, almost every access hits. A factor of eight above it, almost every access misses. The transition is not gradual and it is not a property of any algorithm — it is a property of how much data there is, and an algorithm's complexity class says nothing about which side of it a program is working on.
The floor moves when the question does
Sorting 4,096 elements needs at least 43,250 comparisons. Finding one element among the same 4,096, already sorted, needs at least 13. The difference is a factor of 3,300 and it comes entirely from how many different answers the algorithm has to be able to give. A lower bound is a property of the question, not of any algorithm.
Where an algorithm looks
Plotted as index against time, every array access an algorithm makes becomes a picture that no count contains. Merge sort's is a set of sweeps. Heapsort's is a spray. Quicksort's is a narrowing triangle. These shapes decide how fast the algorithms run and they are entirely absent from the analysis that says all three are Θ(n log n).
A triangle stored in a square
An interval table has a cell for every range of keys and nothing below its diagonal, and it can be stored as a square array, as packed rows, or as packed diagonals — the last matching the order it is filled in. On sixty-four keys, with every read replayed through a small cache, the square misses 39.7% of its reads, packed rows 38.8%, and packed diagonals 78.4%. Storing a table in the order it is written is storing it in the order it is not read.
The permutation that moves almost nothing
Two ways to scramble sixteen thousand elements. Shuffling them inside windows of five hundred and twelve puts two million pairs out of order and costs 3,095 block transfers to carry out. Swapping a thousand pairs across the whole array puts seven million out of order and costs 1,189. Inversions are the textbook measure of disorder, and on a disk they rank these two backwards.
The keys that arrive late
Insert 131,072 keys into a B+-tree in random order and its leaves end up 70.5% full; in ascending order, 51.6%; in descending order, 50.0%. The rule databases use to fix ascending inserts — split a full leaf at its right-hand end — fills them completely, and it does nothing for descending keys. Let one key in a hundred arrive late in an otherwise ascending stream and the rule's leaves fall from 100% to 53.4% full. How much of an index is empty is decided by the order its keys arrived in, and a trickle of disorder undoes the fix.
The order with the best depth
An edit-distance table can be filled row by row, column by column, or one anti-diagonal at a time, and the anti-diagonal order is the one that needs the fewest rounds — 513 against 65,793 on two strings of 256 characters, because every cell on an anti-diagonal is independent of the others. Stored the usual way, row by row, it also misses the cache on 31.1% of its reads, where row order misses 6.3%. The order that is best for parallel work is worst for the memory it runs on.
The bucket that fits a line
Make each of a cuckoo table's two candidates a bucket of eight slots laid out on one cache line, and no lookup ever touches more than two lines, the table builds past a load of 0.95, and at that load it misses 1.21 times a lookup where linear probing misses 1.79. The prediction that it would lose to linear probing at low loads was wrong — it misses less at every load measured, 0.94 against 0.96 at 0.3 — because a key it holds almost never lives in its second bucket. The guarantee belongs to the alignment, not the bucket; eight slots on lines of four put a lookup on four lines.
The table stored the way it is filled
Store an edit-distance table by anti-diagonals instead of by rows, and the anti-diagonal fill keeps its 513 rounds while its cache misses fall from 31.1% of reads to 8.7%. It does not fall to row order's 6.3%, and the gap is not noise — on caches of four and eight lines the two rates are 9.4% and 6.3%, exactly three to two, because a cell reads from two earlier diagonals and only one earlier row. The same layout turns row order into the order that strides, at 28.3%. How a table is stored and the order it is filled in are one decision, and its price is the number of earlier fronts the recurrence reads.
Named alongside it
The objects these essays reach for when they reach for this one.
CacheLocalityWorking setMiss rateMemory layoutTrade offEvaluation orderQuicksortBlock transferComparison countCompulsory missDepth