Perfect Sorting by Reversals Is Not Always Difficult

Minimum Common String Partition Problem: Hardness and Approximations

The Electronic Journal of Combinatorics ◽

10.37236/1947 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 12

Author(s):

Avraham Goldstein ◽

Petr Kolman ◽

Jie Zheng

Keyword(s):

Genome Rearrangement ◽

Linear Time ◽

Fundamental Problem ◽

Text Processing ◽

Partition Problem ◽

Sorting By Reversals ◽

String Comparison ◽

Minimum Number ◽

Tight Connection ◽

Minimum Common String Partition

String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing and compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement. A partition of a string $A$ is a sequence ${\cal P} = (P_1,P_2,\dots,P_m)$ of strings, called the blocks, whose concatenation is equal to $A$. Given a partition ${\cal P}$ of a string $A$ and a partition ${\cal Q}$ of a string $B$, we say that the pair $\langle{{\cal P},{\cal Q}}\rangle$ is a common partition of $A$ and $B$ if ${\cal Q}$ is a permutation of ${\cal P}$. The minimum common string partition problem (MCSP) is to find a common partition of two strings $A$ and $B$ with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most $k$ times in each input string, is denoted by $k$-MCSP. In this paper, we show that $2$-MCSP (and therefore MCSP) is NP-hard and, moreover, even APX-hard. We describe a $1.1037$-approximation for $2$-MCSP and a linear time $4$-approximation algorithm for $3$-MCSP. We are not aware of any better approximations.

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Sorting by Reversals: Genome Rearrangements in Plant Organelles and Evolutionary History of X Chromosome

Molecular Biology and Evolution ◽

10.1093/oxfordjournals.molbev.a040208 ◽

1995 ◽

Keyword(s):

X Chromosome ◽

Evolutionary History ◽

Genome Rearrangements ◽

Sorting By Reversals ◽

History Of ◽

Plant Organelles

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AVERAGE-CASE ANALYSIS OF PERFECT SORTING BY REVERSALS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001280 ◽

2011 ◽

Vol 03 (03) ◽

pp. 369-392 ◽

Cited By ~ 2

Author(s):

MATHILDE BOUVEL ◽

CEDRIC CHAUVE ◽

MARNI MISHNA ◽

DOMINIQUE ROSSIN

Keyword(s):

Average Length ◽

Algorithm Analysis ◽

Sorting Algorithm ◽

Average Case Analysis ◽

Signed Permutation ◽

Average Case ◽

Combinatorial Properties ◽

Sorting By Reversals ◽

Identity Permutation ◽

Interval Trees

Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. Bérard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, polynomial, and additionally, for sufficiently long permutations, the sorting algorithm runs in polynomial time with probability one. Furthermore, our analysis of the subclass of commuting scenarios yields precise results on the average length of a reversal, and the average number of reversals.

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