Pancake Flipping with Two Spatulas

TL;DR

This paper explores various algorithms for sorting permutations using prefix operations, improving approximation ratios and analyzing practical performance, with applications in networks and biology.

Contribution

It introduces new approximation algorithms for sorting permutations with prefix reversals, transpositions, and transreversals, and analyzes their practical effectiveness.

Findings

3-approximation algorithm for sorting with prefix reversals and transpositions

2-approximation algorithm for sorting with prefix reversals and transreversals

Approximation ratios improve with more prefix reversals in optimal solutions

Abstract

In this paper we study several variations of the \emph{pancake flipping problem}, which is also well known as the problem of \emph{sorting by prefix reversals}. We consider the variations in the sorting process by adding with prefix reversals other similar operations such as prefix transpositions and prefix transreversals. These type of sorting problems have applications in interconnection networks and computational biology. We first study the problem of sorting unsigned permutations by prefix reversals and prefix transpositions and present a 3-approximation algorithm for this problem. Then we give a 2-approximation algorithm for sorting by prefix reversals and prefix transreversals. We also provide a 3-approximation algorithm for sorting by prefix reversals and prefix transpositions where the operations are always applied at the unsorted suffix of the permutation. We further analyze the problem in more practical way and show quantitatively how approximation ratios of our algorithms improve with the increase of number of prefix reversals applied by optimal algorithms. Finally, we present experimental results to support our analysis.