Approximation algorithms for sorting permutations by extreme block-interchanges

Abstract

Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two substrings that are called as blocks in literature, in a given permutation. When the blocks are restricted to be adjacent then one obtains a well studied operation: transposition. We call either a prefix or a suffix as an extreme. Restricting one of the swapped blocks to be an extreme in block-interchange operation yields a prefix or a suffix block-interchange respectively, the two types of extreme block-interchanges. For prefix block-interchange operation over permutations we design: (i) an optimum algorithm to sort reverse permutation, Rn, in n/2 moves, (ii) a simple 2-approximation algorithm, and (iii) for permutations with O(1) cycles, a 4/3 approximation algorithm. Due to symmetry, these results apply to suffix block-interchange operation also.