Randomization function: Difference between revisions

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In [[computer science]], '''randomization function''' or '''randomizing function''' is a [[algorithm]] or [[procedure]] that implements a [[random]]ly chosen [[function (mathematics)|function]] between two specific [[set (mathematics)|set]]s, suitable for use in a [[randomized algorithm]].

Randomizing functions are centralused to the design of [[randomizedturn algorithms]], that have good [[expected value (statistics)|expected]] performance for ''random'' inputs, into algorithms that have the same performance for ''any'' input. For example, consider ana [[sorting algorithm]] like [[quicksort]], which has small expected running time forwhen the input items are presented in random inputsorder, but is very slow when the input datathey are presented in certain unfavorable orders. A randomizing function from the integers 1 to ''n'' to the integers 1 to ''n'' can be used used to rerrange the ''n'' input items in "random" order, before calling that algorithm. This modified (randomized) algorithm will have small expected running time, forwhatever ''any''the input order.

Randomizing functions are related to [[random number generator]]s and [[hash function]]s, but have somewhat different requirements and uses, and often need specific algorithms.

==Requirements==

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===Randomness===

In theory, randomization functions are assumed to be truly random, and yield an unpredictably different function every time the algorithm is executed. The randomization technique would not work if, at every execution of the algorithm, the randomization function always performed the same mapping, or a mapping entirely determined by some externally observable parameter (such as the program's startup time). With such a "pseudo-randomization" function, one could in principle construct a sequence of calls such that the function would always yield a "bad" case for the underlying deterministic algorithm. For that sequence of calls, the average cost would be closer to the worst-case cost, rather than the average cost for random inputs.

In practice, however, the main concern is that some bad cases for the deterministic algorithm may occur in practice much more often than it would be predicted by chance. For example, in a naive variant of quicksort, the worst case is when the input items are already sorted — which is a very common occurrence in many appilcations. For such algorithms, even a fixed pseudo-random premutation may be good enough. Even though the resulting "pseudo-randomized" algorithm would still have some "bad" cases, they will be certain peculiar orders that would be quite unlikely to arise in real applications. So, in practice one often uses functions that are derived from [[pseudo-random number generators]], preferably [[random seed|seeded]] with external "random" data such as the program's startup time.

===Uniformity===

The uniformiy requirements for a randomizing function are usually much weaker than those of hash functions and pseudo-random generators. The minimum requirement is that they map any input of the deterministic algorithm into a "good" input with a sufficiently hight probability. (However, analysis is usuallu simpler if the randomizing function implements each possible mapping with uniform probability.)

==References==

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