The Greedy Algorithm for the Minimum Common String Partition Problem

Chrobák, Marek; Kolman, Petr; Sgall, Jiřį́

doi:10.1007/978-3-540-27821-4_8

Cited by 44 publications

(43 citation statements)

References 5 publications

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“…Christie and Irving [4] prove that unsigned SBR is NP-hard for binary strings and Chen et al [3] show that 2-SBR is NP-hard. The best approximation ratio for the general signed SBR is O(log n log * n) (following from the work of Cormode and Muthukrishnan [6]); there are O(1)-approximation algorithms for signed 2-SBR and 3-SBR [3,5,9]. Kolman [11] describes a greedy-like O(k 2 )-approximation algorithm for k-SBR running in O(kn) time.…”

Section: Introductionmentioning

confidence: 99%

Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set

Kolman¹,

Waleń²

2007

Electron. J. Combin.

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In the last decade there has been an ongoing interest in string comparison problems; to a large extend the interest was stimulated by genome rearrangement problems in computational biology but related problems appear in many other areas of computer science. Particular attention has been given to the problem of sorting by reversals (SBR): given two strings, $A$ and $B$, find the minimum number of reversals that transform the string $A$ into the string $B$ (a reversal $\rho(i,j)$, $i < j$, transforms a string $A=a_1\ldots a_n$ into a string $A'=a_1\ldots a_{i-1} a_{j} a_{j-1} \ldots a_{i} a_{j+1} \ldots a_n$). Closely related is the minimum common string partition problem (MCSP): given two strings, $A$ and $B$, find a minimum size partition of $A$ into substrings $P_1,\ldots,P_l$ (i.e., $A=P_1\ldots P_l$) and a partition of $B$ into substrings $Q_1,\ldots,Q_l$ such that $(Q_1,\ldots,Q_l)$ is a permutation of $(P_1,\ldots,P_l)$. Primarily the SBR problem has been studied for strings in which every symbol appears exactly once (that is, for permutations) and only recently attention has been given to the general case where duplicates of the symbols are allowed. In this paper we consider the problem $k$-SBR, a version of SBR in which each symbol is allowed to appear up to $k$ times in each string, for some $k\geq 1$. The main result of the paper is a $\Theta(k)$-approximation algorithm for $k$-SBR running in time $O(n)$; compared to the previously known algorithm for $k$-SBR, this is an improvement by a factor of $\Theta(k)$ in the approximation ratio, and by a factor of $\Theta(k)$ in the running time. We approach the $k$-SBR by finding an approximation for the $k$-MCSP first and then turning it into a solution for $k$-SBR. Crucial ingredients of our algorithm are the suffix tree data structure and a linear time algorithm for a special case of a disjoint set union problem.

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Section: Introductionmentioning

confidence: 99%

Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set

Kolman¹,

Waleń²

2007

Electron. J. Combin.

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“…The traditional edit distance is then applied on the new strings. Chrobak et al [12] consider the Minimum Common String Partition (MCSP) problem, that receives two input strings and tries to minimize the number of partitions of the strings into the same collection of substrings. They refer to several versions of this problem by limiting the number of times each character can occur in both input strings, and study the approximation of a greedy algorithm for MCSP that at each step extracts a longest common substring from the given strings.…”

Section: Related Workmentioning

confidence: 99%

Edit Distance with Block Deletions

Shapira

Storer

2011

Algorithms

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Several variants of the edit distance problem with block deletions are considered. Polynomial time optimal algorithms are presented for the edit distance with block deletions allowing character insertions and character moves, but without block moves. We show that the edit distance with block moves and block deletions is NP-complete (Nondeterministic Polynomial time problems in which any given solution to such problem can be verified in polynomial time, and any NP problem can be converted into it in polynomial time), and that it can be reduced to the problem of non-recursive block moves and block deletions within a constant factor

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“…The hardness results from above led to the development of approximation algorithms for move operations. The results include a greedy algorithm presented in Shapira and Storer (2002) which achieves an approximation factor of O(n 0.69 ) as shown by Chrobak et al (2005). Furthermore, Cormode and Muthukrishnan (2002) presented a O(log * n log n) factor approximation algorithm for general move operations which runs in sub-quadratic time.…”

Section: Introductionmentioning

confidence: 95%

Alignments with non-overlapping moves, inversions and tandem duplications in O(n 4) time

Ledergerber

Dessimoz

2007

J Comb Optim

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Sequence alignment is a central problem in bioinformatics. The classical dynamic programming algorithm aligns two sequences by optimizing over possible insertions, deletions and substitutions. However, other evolutionary events can be observed, such as inversions, tandem duplications or moves (transpositions). It has been established that the extension of the problem to move operations is NP-complete. Previous work has shown that an extension restricted to non-overlapping inversions can be solved in O(n 3) with a restricted scoring scheme. In this paper, we show that the alignment problem extended to non-overlapping moves can be solved in O(n 5) for general scoring schemes, O(n 4 log n) for concave scoring schemes and O(n 4) for restricted scoring schemes. Furthermore, we show that the alignment problem extended to non-overlapping moves, inversions and tandem duplications can be solved with the same time complexities. Finally, an example of an alignment with non-overlapping moves is provided.

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The Greedy Algorithm for the Minimum Common String Partition Problem

Cited by 44 publications

References 5 publications

Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set

Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set

Edit Distance with Block Deletions

Alignments with non-overlapping moves, inversions and tandem duplications in O(n 4) time

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