Strictly In-Place Algorithms for Permuting and Inverting Permutations

TL;DR

This paper introduces new strictly in-place algorithms for permuting and inverting permutations with near-linear time complexity, improving upon previous methods that used more space or had higher time complexity.

Contribution

It presents the first strictly in-place algorithms for permutation and inversion that run in near-linear time, generalizing previous methods and surpassing recent results.

Findings

Achieved $ ilde{O}(n)$ time complexity for permutation and inversion algorithms.

Extended the method of Fich et al. to strictly in-place algorithms.

Improved the time complexity over recent algorithms by Guépil.

Abstract

We revisit the problem of permuting an array of length $n$ according to a given permutation in place, that is, using only a small number of bits of extra storage. Fich, Munro and Poblete [FOCS 1990, SICOMP 1995] obtained an elegant $\mathcal{O}(n\log n)$-time algorithm using only $\mathcal{O}(\log^{2}n)$ bits of extra space for this basic problem by designing a procedure that scans the permutation and outputs exactly one element from each of its cycles. However, in the strict sense in place should be understood as using only an asymptotically optimal $\mathcal{O}(\log n)$ bits of extra space, or storing a constant number of indices. The problem of permuting in this version is, in fact, a well-known interview question, with the expected solution being a quadratic-time algorithm. Surprisingly, no faster algorithm seems to be known in the literature. Our first contribution is a strictly in-place generalisation of the method of Fich et al. that works in $\mathcal{O}_{\varepsilon}(n^{1+\varepsilon})$ time, for any $\varepsilon > 0$. Then, we build on this generalisation to obtain a strictly in-place algorithm for inverting a given permutation on $n$ elements working in the same complexity. This is a significant improvement on a recent result of Gu\'spiel [arXiv 2019], who designed an $\mathcal{O}(n^{1.5})$-time algorithm.