M. Frances and A. Litman. On covering problems of codes. Theor. Comput. Syst., 30:113--119, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# # d # l. Parameter: Alphabet # and positiveintegers m; p; n; l and d. Question: Is there a subset # of strings in # l such that for each string S ##, there is a string c ##that has Hamming distance # d from some length l substring of # When l # n and p ##, p## is known as ######## ###### [6] and has been investigated in the context of coding theory. When l # n and p # #, p## is known as ####### ###### p########### [8] and has been investigated in the context of string clustering. When l # n and p ##, p## is known as ####### ######### [12,13]andhasbeeninvestigated in the context of ....

....describe the input size of an instance, we analyse it as a parameter for the purpose of completeness. Section 5 lists some promising directions for future research. 2 ########### ######### This problem is also known as ####### ######### [12,13] It is NP hard by results derived independently in [6] and [12] for ####### ######, i.e. ########### ######### when l # n. Several polynomial time approximation algorithms that give solutions within a multiplicative factor of 2 of the optimal value of d are known [12, 13] and a polynomial time approximation scheme (PTAS) has also recently been ....

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M. Frances and A. Litman. 1997. On Covering Problems of Codes. Theory of Computing Systems, 30, 113119.

.... problems find applications in locating binding sites and finding conserved regions in unaligned sequences [24, 11, 10, 23] genetic drug target identification [13] designing genetic probes [13] universal PCR primer design [12, 4, 21, 13] and, outside computational biology, in coding theory [5, 6]. Such problems may be considered to be various generalizations of the common substring problem, allowing errors. Many measures have been proposed for finding such regions common to every given string. A popular and most fundamental measure is the Hamming distance. Moreover, two popular objective ....

....t i ) d. Throughout this paper, we call the number d in the definitions of Closest String and Closest Substring the radius (or the cost) of the solution. Closest String has been widely and independently studied in different contexts. In the context of coding theory it was shown to be NP hard [5]. In DNA sequence related topics, 3] gave an exact algorithm when the distance d is a constant. 2, 6] gave near optimal approximation algorithms only for large d (super logarithmic in number of sequences) however the straightforward LP (linear programming) relaxation technique does not work ....

[Article contains additional citation context not shown here]

M. Frances, A. Litman, On covering problems of codes, Theor. Comput. Syst., 30, 113-119, 1997.

....measure of [26, 16, 12, 25] As an interesting application of our analysis, we further obtain a PTAS for a restricted (but still NP hard) version of the important star alignment problem allowing at most constant number of gaps, each of arbitrary length, in each sequence. The Closest String problem [2, 3, 7, 9, 18] asks for the smallest d and a string s which is within Hamming distance d to each s i . The problem is NP hard [7, 18] 3] gives a polynomial time algorithm for constant d. For super logarithmic d, 2, 9] give efficient approximation algorithms using linear program ralaxation techniques. The ....

....NP hard) version of the important star alignment problem allowing at most constant number of gaps, each of arbitrary length, in each sequence. The Closest String problem [2, 3, 7, 9, 18] asks for the smallest d and a string s which is within Hamming distance d to each s i . The problem is NP hard [7, 18]. 3] gives a polynomial time algorithm for constant d. For super logarithmic d, 2, 9] give efficient approximation algorithms using linear program ralaxation techniques. The best polynomial time approximation has ratio 4 3 for all d, given by [18] 9] also independently claimed the 4 3 ratio ....

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M. Frances, A. Litman, On covering problems of codes, Theor. Comput. Syst. 30(1997) 113-119.

....it should be noted though, that the problem of nding the longest almost common substring for general c is NP hard when using the Hamming distance. This can be shown by an easy reduction from the problem of determining the consensus that has been shown NP complete by Frances and Litman in [18]. The consensus for a set of strings S = fs 1 ; s k g is a string x, that minimises max s i 2S d(x; s i ) Determining the consensus via the almost common substring can be done by a (binary) search for the smallest number of changes c, that have to be allowed in order for the almost common ....

M. Frances and A. Litman. On covering problems of codes. Technical Report 827, Technion, Israel, jul 1994.

....DNA) to bind tightly to another strand, one may use Hamming distance, which considers only substitutions, rather than edit distance, which considers both gaps and substitutions. 2 Applications The Hamming distance metric appears in several contexts. In particular, it is used in coding theory [FL97, GJL99], and in several biological applications. The biological applications occur in two varieties: some require that a region of similarity be discovered, for example consensus sequences, and other applications use the reverse complement of that region, such as designing probes or primers. Our ....

....distance and m is the number of strings. However, a small d is critical for our applications and the straightforward LP relaxation method as used in [BL 97] does not work well for small d s. The Closest String Problem also occurs in coding theory, and has been proven NP Complete for binary codes [FL97]. Again in the context of coding theory, G asieniec et al. GJL99] independently claim a weaker ( 4 3 ffl) approximation, with the requirement that d is large (super logarithmic) Our ( 4 3 ffl) approximation in this paper works for all d. 5 The Complexity of Farthest String Problem In ....

Frances, M., and Litman, A. (1997) On Covering Problems of Codes, Theory of Computing Systems 30, 113-119.

....Jesper Jansson y Andrzej Lingas y Abstract The Hamming center problem for a set S of k binary strings, each of length n, is to find a binary string fi of length n that minimizes the maximum Hamming distance between fi and any string in S. Its decision version is known to be NP complete [2]. We provide several approximation algorithms for the Hamming center problem. Our main result is a randomized ( 4 3 ) approximation algorithm running in polynomial time if the Hamming radius of S is at least superlogarithmic in k. Furthermore, we show how to find in polynomial time a set B ....

....f Jesper.Jansson, Andrzej.Lingas g dna.lth.se straightforward polynomial time algorithm for HCP as long as the radius is O(1) see Section 4) Berman et al. left the question open whether there exists an efficient algorithm for the case where the size of the radius is arbitrarily large. In [2], Frances and Litman showed that the decision version of HCP, by them called the Minimum Radius Problem, as well as the equivalent dual Maximum Covering Radius Problem (MCR) are NP complete. Motivated by the intractability of HCP, we present several approximation algorithms. Some similar results ....

M. Frances and A. Litman, On Covering Problems of Codes, Theory of Comp. Syst. 30, pp. 113--119, 1997.

....measure of [26, 16, 12, 25] As an interesting application of our analysis, we further obtain a PTAS for a restricted (but still NP hard) version of the important star alignment problem allowing at most constant number of gaps, each of arbitrary length, in each sequence. The Closest String problem [2, 3, 7, 9, 18] asks for the smallest d and a string s which is within Hamming distance d to each s i . The problem is NP hard [7, 18] 3] gives a polynomial time algorithm for constant d. 2, 9] give efficient approximation algorithms for super logarithmic d, using linear program ralaxation techniques. The ....

....NP hard) version of the important star alignment problem allowing at most constant number of gaps, each of arbitrary length, in each sequence. The Closest String problem [2, 3, 7, 9, 18] asks for the smallest d and a string s which is within Hamming distance d to each s i . The problem is NP hard [7, 18]. 3] gives a polynomial time algorithm for constant d. 2, 9] give efficient approximation algorithms for super logarithmic d, using linear program ralaxation techniques. The best polynomial time approximation has ratio 4 3 for all d, given by [18] 9] also independently claimed the 4 3 ....

[Article contains additional citation context not shown here]

M. Frances, A. Litman, On covering problems of codes, Theor. Comput. Syst. 30(1997) 113-119.

....distance and m is the number of strings. However, a small d is critical for our applications and the straightforward LP relaxation method as used in [BL 97] does not work well for small d s. The Closest String Problem also occurs in coding theory, and has been proven NP Complete for binary codes [FL97]. Again in the context of coding theory, G asieniec et al. GJL99] present a similar 4 3 approximation, but with the requirement that d is large (super logarithmic) Our 4 3 approximation in this paper works for all d. 5 The Complexity of Farthest String Problem In this section, we prove ....

Frances, M., and Litman, A. (1997) On Covering Problems of Codes, Theory of Computing Systems, 30, 113-119.

.... then the following: given a set of sequences V Sigma n , representing the rows of some multiple alignment, find a sequence c 2 Sigma n (the consensus) such that R(V; c) R(V ) in other words, find the sequence c which minimizes max a2V d(c; a) It has been shown by Frances and Litman ([FL94]) that determining the consensus is NP complete (using a reduction from 3 SAT) in the special case where the alphabet is binary and the distance measure is the Hamming distance. A slight modification of their reduction generalizes the hardness result to arbitrary finite alphabet Sigma . Assuming ....

M. Frances and A. Litman. On covering problems of codes. Technical Report 827, Technion, Israel, July 1994.

.... in Ntafos and Hakimi [124] The NP completeness of deciding whether a binary code has a vector of weight n=2 has been proved by Calderbank and Shor (see [47] The same result for ternary codes and weight n is from Barg [19] For the NP completeness of P 5 ; P 6 ; P 7 see Frances and Litman [65], Horn and Kschischang [84] and Kratochvil [100] respectively. An easy reduction for P 6 follows from part (b) of Theorem 4.1; see Vardy [161] The construction of codes from graphs was introduced in Calabi [34] Hakimi [81] Properties of classical linear spaces in graphs and their ....

M. Frances and A. Litman, "On covering problems of codes," Theory of Computing Systems, 30 (2) (1997), 113--119.

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M. Frances and A. Litman. On covering problems of codes. Theor. Comput. Syst., 30:113--119, 1997.

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Frances, M. and Litman, A. 1997. On Covering Problems of Codes. Theory of Computing Systems 30:113-119.

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