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The sequence of permutations generated by the Steinhaus–Johnson–Trotter algorithm has a natural recursive structure, that can be generated by a recursive algorithm. . However the actual Steinhaus–Johnson–Trotter algorithm does not use recursion, instead computing the same sequence of permutations by a simple iterative me
In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer. [2] The sequence of permutations of n objects generated by Heap's algorithm is the beginning of the sequence of permutations of n+1 objects.
The algorithm produces an unbiased permutation: every permutation is equally likely. The modern version of the algorithm takes time proportional to the number of items being shuffled and shuffles them in place. The Fisher–Yates shuffle is named after Ronald Fisher and Frank Yates, who first described it.
The run time grows to O(nlog(n)) if all elements must be distinct. Bogosort has O(n) time when the elements are sorted on the first iteration. In each iteration all elements are checked if in order. There are n! possible permutations; with a balanced random number generator, almost each permutation of the array is yielded in n! iterations.
An algorithm is said to be constant time (also written as () time) if the value of () (the complexity of the algorithm) is bounded by a value that does not depend on the size of the input. For example, accessing any single element in an array takes constant time as only one operation has to be performed to locate it.
Knuth characterized the permutations that this algorithm correctly sorts as being exactly the permutations that do not contain the permutation pattern 231: three elements x, y, and z, appearing in the input in that respective order, with z < x < y. Moreover, he observed that, if the algorithm fails to sort an input, then that input cannot be ...
In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size n is in NC if there exist constants c and k such that it can be solved in time O ((log n ) c ) using O ...
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.