Adjacent Swaps on Strings

Abstract

Transforming strings by exchanging elements at bounded distance is applicable in fields like molecular biology, pattern recognition and music theory. A reversal of length two at position i is denoted by (i i+1). When it is applied to π, where π = π ₁,π ₂, π ₃,..., π _i ,π _i + 1, π _n , it transforms π to π′, where π′ = π ₁,π ₂, π ₃,..., π _i − 1,π _i + 1, π _i , π _i + 1, ..., π _n . We call this operation an adjacent swap. We study the problem of computing the minimum number of adjacent swaps needed to transform one string of size n into another compatible string over an alphabet σ of size k, i.e. adjacent swap distance problem. O(nlog ₂ n) time complexity algorithms are known for adjacent swap distance. We give an algorithm with O(nk) time for both signed and unsigned versions of this problem where k is the number of symbols. We also give an algorithm with O(nk) time for transforming signed strings with reversals of length up to 2, i.e. reversals of length 1 or 2.