In-situ associative permuting

TL;DR

This paper introduces in-situ associative permuting, a space-efficient method for permuting and sorting integer arrays in-place with minimal extra space, combining permuting and inverting techniques.

Contribution

It presents a novel in-situ associative permuting technique that efficiently permutes and sorts integer arrays using only logarithmic additional bits.

Findings

Permutes array elements in O(n) time with logn bits of extra space.

Transforms keys into associatively permutable permutations.

Enables in-place sorting of integer keys with minimal auxiliary space.

Abstract

The technique of in-situ associative permuting is introduced which is an association of in-situ permuting and in-situ inverting. It is suitable for associatively permutable permutations of {1,2,...,n} where the elements that will be inverted are negative and stored in order relative to each other according to their absolute values. Let K[1...n] be an array of n integer keys each in the range [1,n], and it is allowed to modify the keys in the range [-n,n]. If the integer keys are rearranged such that one of each distinct key having the value i is moved to the i'th position of K, then the resulting arrangement (will be denoted by K^P) can be transformed in-situ into associatively permutable permutation pi^P using only logn additional bits. The associatively permutable permutation pi^P not only stores the ranks of the keys of K^P but also uniquely represents K^P. Restoring the keys from pi^P is not considered. However, in-situ associative permuting pi^P in O(n) time using logn additional bits rearranges the elements of pi^P in order, as well as lets to restore the keys of K^P in O(n) further time using the inverses of the negative ranks. This means that an array of n integer keys each in the range [1,n] can be sorted using only logn bits of additional space.