Prefix reversals on binary and ternary strings

C.A.J. Hurkens, L.J.J. van Iersel, J.C.M. Keijsper, S.M. Kelk, L. Stougie, J.T. Tromp

    Research output: Contribution to JournalArticleAcademicpeer-review

    Abstract

    Given a permutation $\pi$, the application of prefix reversal $f^{(i)}$ to $\pi$ reverses the order of the first $i$ elements of $\pi$. The problem of sorting by prefix reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Discrete Math., 27 (1979), pp. 47–57), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings; and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed
    Original languageEnglish
    Pages (from-to)592-611
    JournalSIAM Journal on Discrete Mathematics
    Volume21
    Issue number3
    DOIs
    Publication statusPublished - 2007

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