Worst-case complexity Information

From Wikipedia
https://en.wikipedia.org/wiki/Worst-case_complexity

In computer science (specifically computational complexity theory), the worst-case complexity measures the resources (e.g. running time, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as n in asymptotic notation). It gives an upper bound on the resources required by the algorithm.

In the case of running time, the worst-case time complexity indicates the longest running time performed by an algorithm given any input of size n, and thus guarantees that the algorithm will finish in the indicated period of time. The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms.

The worst-case complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses on a random input.

Definition

Given a model of computation and an algorithm that halts on each input , the mapping is called the time complexity of if, for every input string , halts after exactly steps.

Since we usually are interested in the dependence of the time complexity on different input lengths, abusing terminology, the time complexity is sometimes referred to the mapping , defined by the maximal complexity

of inputs with length or size .

Similar definitions can be given for space complexity, randomness complexity, etc.

Ways of speaking

Very frequently, the complexity of an algorithm is given in asymptotic Big-O Notation, which gives its growth rate in the form with a certain real valued comparison function and the meaning:

  • There exists a positive real number and a natural number such that

Quite frequently, the wording is:

  • „Algorithm has the worst-case complexity .“

or even only:

  • „Algorithm has complexity .“

Examples

Consider performing insertion sort on numbers on a random access machine. The best-case for the algorithm is when the numbers are already sorted, which takes steps to perform the task. However, the input in the worst-case for the algorithm is when the numbers are reverse sorted and it takes steps to sort them; therefore the worst-case time-complexity of insertion sort is of .

See also

References

  • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN  0-262-03293-7. Chapter 2.2: Analyzing algorithms, pp.25-27.
  • Oded Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, 2008. ISBN  0-521-88473-X, p.32.