The cost of cyclic permutations and remainder sums in the Euclidean algorithm

TL;DR

This paper analyzes the average and worst-case costs of a modified in-place rotation algorithm, linking it to the asymptotic behavior of remainders in the Euclidean algorithm, and provides cost bounds.

Contribution

It introduces a modification to the Gries-Mills block swapping scheme and analyzes its cost using the Euclidean algorithm's properties.

Findings

Average cost of 1.85 moves per element

Worst case cost remains at 3 moves per element

Analysis based on the asymptotic behavior of remainders

Abstract

We discuss a modification to the Gries-Mills block swapping scheme for in-place rotation with average costs of 1.85 moves per element and worst case performance still at 3 moves per element. Analysis of the average case relies on the asymptotic behavior of the sum of remainders in the Euclidean algorithm.