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

Asymptotic notation — where it appears

A statement about how a cost grows as its parameter grows without bound, with constants and lower-order terms dropped. The dropped term is frequently the majority of the quantity at any size anybody runs, which is what makes a measurement necessary beside it.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

10010³10010³10⁴10⁵10⁶ncomparisonsInsertionHeapsortMergeShellsorta power law is a straight line herecomparisons, counted exactly

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.

bounds · Bound sampled positions3,855grows with nsample marks2,056grows with nrun starts2,016grows with rrun lengths per symbol1,885grows with rrun heads1,563grows with rC table525grows with r11,900 bits in total16,384 characters · r = 224 · sample one in 6450% is the sampling

What is still proportional to n

An index whose size is a function of the run count still has an n in it, and at thirty-two copies of a text the n is half of it. Everything that stores the transform grew by 45 per cent; the two arrays that answer "where" grew by eighteen times, and neither of them has anything to do with repetition.

space · Repeat 10010³10⁴10⁵errors allowed, kacts01234index walkthe whole table4,000 characters · m = 16 · 4 symbolscrossing at k = 4

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.

bounds · Distance the same 16 patterns, one of them shortenedshortest 1658.4%mean shift 3.15shortest 1262.8%mean shift 2.99shortest 869.7%mean shift 2.64shortest 680.5%mean shift 2.35shortest 493.4%mean shift 1.98shortest 399.2%mean shift 1.71shortest 2100.0%mean shift 1.3716 patterns, 15 of them 16 characterstext 20,000

The ceiling the shortest pattern sets

A matcher that skips is described as faster than one that reads every character, and the description leaves out what decides it. No shift can exceed the shortest pattern in the set, so adding one two-character pattern to fifteen of sixteen characters takes a run from reading fifty-eight per cent of the text to reading all of it twice.

wrong · Symbol

Named alongside it

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

MeasurementConstant factorUpper boundAho-corasickAlphabetAlphabet sizeApproximate matchingBacktracking searchBackward searchBranching factorComparison countCounterexample

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