Approximate Block Sorting

Abstract

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n3) time 2-approximation algorithm for BLOCK-SORTING. We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

A quadratic time 2-approximation algorithm for block sorting

“…The block merging problem of Mahajan et al [14] is defined as follows: Given a permutation, at each step a configuration of the problem consists of a set of sorted lists of integers, where each integer is in exactly one list. One or more of the lists may be empty.…”

“…We mention work on sorting with prefix reversals [5,7,11], transpositions [1,2,6], and block moves [9,10,[14][15][16][17]. In particular, block sorting is not the same as transposition sorting, and, thus, the 1.5-approximation to optimality obtained by Bafna and Pevzner [1,2] or the 1.375-approximation of Elias and Hartman [6] does not imply a 1.5-approximation to optimum block sorting.…”

“…The first approximation algorithm with a non-trivial approximation ratio for the block sorting problem was given in [14,15] by Mahajan, Rama, and Vijayakumar. They gave an O(n 3 )-time block sorting algorithm with approximation ratio 2.…”

Algorithms and Design for an Autonomous Biological System

“…Some well known primitives for genome rearrangements are transpositions [5], reversals [3], [4], and strip moves [1], [2], [6]. A reversal inverts the order of a substring of any length, a transposition swaps two adjacent substrings of any length without changing the order of two substrings, and a strip move moves a strip to any other location in a permutation.…”

“…Sorting by strip moves is motivated by applications in Optical Character Recognition (OCR) [1], [2], [6]. Text regions referred to as zones are selected by drawing rectangles around them.…”