Bogosort

Class Sorting algorithm Array Unbounded[1] Ω(n)[1] O(n × n!)[1] O(n)

In computer science, bogosort[1][2] (also stupid sort,[3] slowsort,[4][5] random sort, shotgun sort or monkey sort [6]) is a particularly ineffective sorting algorithm based on the generate and test paradigm. It is not useful for sorting, but may be used for educational purposes, to contrast it with other more realistic algorithms; it has also been used as an example in logic programming.[2][4][5] If bogosort were used to sort a deck of cards, it would consist of checking if the deck were in order, and if it were not, throwing the deck into the air, picking the cards up at random, and repeating the process until the deck is sorted. Its name comes from the word bogus.[7]

Description of the algorithm

The following is a description of the algorithm in pseudocode:

while not isInOrder(deck):
shuffle(deck)


Running time and termination

This sorting algorithm is probabilistic in nature. If all elements to be sorted are distinct, the expected number of comparisons in the average case is asymptotically equivalent to $(e-1) n!$, and the expected number of swaps in the average case equals $(n-1) n!$.[1] The expected number of swaps grows faster than the expected number of comparisons, because if the elements are not in order, this will usually be discovered after only a few comparisons no matter how many elements there are, but the work of shuffling the collection is proportional to its size. In the worst case, the number of comparisons and swaps are both unbounded, for the same reason that a tossed coin might turn up heads any number of times in a row.

The best case occurs if the list as given is already sorted; in this case the expected number of comparisons is $n-1$, and no swaps at all are carried out.[1]

For any collection of fixed size, the expected running time of the algorithm is finite for much the same reason that the infinite monkey theorem holds: there is some probability of getting the right permutation, so given an unbounded number of tries it will almost surely eventually be chosen.

Related algorithms

Gorosort
is a sorting algorithm introduced in the 2011 Google Code Jam.[8] As long as the list is not in order, a subset of all elements is randomly permuted. If this subset is optimally chosen each time this is performed, the expected value of the total number of times this operation needs to be done is equal to the number of misplaced elements.
Bogobogosort
is an algorithm that was designed not to succeed before the heat death of the universe on any sizable list. It works by implementing the bogosort on the first two elements in the list. If they are in order, then it bogosorts the first three elements, and so on, increasing by one until the entire list is sorted. Should the list not be in order at any point, the sort starts over with the first two elements.
Bozosort
is another sorting algorithm based on random numbers. If the list is not in order, it picks two items at random and swaps them, then checks to see if the list is sorted. The running time analysis of a bozosort is more difficult, but some estimates are found in H. Gruber's analysis of "perversely awful" randomized sorting algorithms.[1] O(n!) is found to be the expected average case.
Quantum Bogosort
An in-joke among some computer scientists is that quantum computing could be used to effectively implement a bogosort with a time complexity of O(n).[9] First, use quantum randomness to permute the list. The quantum randomization creates n! branches of the wavefunction ("universes" in many-worlds interpretation), and one of these will be such that this single shuffle had produced the list in sorted order. The list is then inspected, and if it is not sorted, the universe is destroyed. Since destroyed universes cannot be observed, the list is always observed to have been successfully sorted after one iteration (having done O(n) work).