Upper bound — where it appears
Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.
What O-notation does not say
Big-O is a statement about a limit. It does not say how fast, it does not say which is better, it does not say anything at all about any particular n, and it discards precisely the factor that usually decides the answer. Knowing exactly what it claims is the difference between using it and being misled by it.
A band as wide as the answer
If two strings are close, the optimal route stays near the diagonal and nine cells in ten cannot be on it. A band of three finds the right answer on a pair 300 characters long — and a band of thirty-two is needed before anything can prove it.
The model a bound was quoted in
Every accuracy figure in this field's first phase was measured under four unstated assumptions. Remove them one at a time and one structure loses its guarantee on 91% of queries, another's error stops falling when it is given more state, and a third has nothing to do at all.
A table wider than its input
The knapsack table has (n+1)(W+1) cells and is called polynomial. Adding one character to the input doubles it — across four settings the table grows sixty-four times while the input it is written from grows by half.
The branches an error opens
The tree multiplies by 19.6 for the first error, 12.3 for the second, 10.3 for the third and 8.8 for the fourth. A branching factor of four on a sixteen-character pattern would predict sixty-four, and the gap between sixty-four and eight is the intervals emptying.
The table that walks every pair
The exact shift rules cost 769,724 character comparisons to build for 128 patterns and the published ones cost 5,604. The scan they are both built for reads 41,580 characters, so one of the two constructions is eighteen times the work it is there to save.
Every substring, in fewer states than substrings
A text of 512 characters has 129,416 distinct substrings. A machine that recognises every one of them, and nothing else, needs 831 states — and the bound it is under, 2n − 1, is reached exactly by a string one line long.
A summary's silence is a bound, not a count
When both sides of a join are skewed towards the same values the textbook estimate is 399 times too low and the order it picks costs 244 times the better one; two counters a side repair it. Skew them towards different values and the same estimate becomes 254 times too high — and the counters have nothing to say, because no value is in both summaries. Using their silence instead repairs the middle of the sweep and makes the far end six times worse.
Named alongside it
The objects these essays reach for when they reach for this one.
Constant factorCost modelApproximate matchingAsymptotic notationCrossoverDynamic programmingEdit distanceGuaranteeLower boundMeasured countMeasurementPruning