Eric Bach, Anne Condon, Elton Glaser, and Celena Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings, Eleventh Annual IEEE Conference on Computational Complexity, pages 290--300, Philadelphia, Pennsylvania, 24--27 May 1996. IEEE Computer Society Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....truth assignments. In this case the number of variables which can be used in a formula is bounded by 34 (since 10 ) The more e#cient heuristic allows us to more than double the number of variables of a formula, and this fact is independent of the current state of technology (see e.g. [3] for algorithms for other NP complete problems) The advantage of DNA computing is the parallelism which can be exploited by the huge number of DNA molecules in a single probe. However, if we implement the trivial approach to solve the 3SAT problem and consider all possible assignments, much of ....

....the number of di#erent extensions of a partial solution, e.g. the number of literals of a clause, or the number of edges leaving a node in a graph problem. The separation into di#erent tubes is necessary since the extension step usually determines some property of the partial solutions (see e.g. [5,3]) In each separate operation there is some not too small probability that the actual solution may be lost. The probability depends on the reaction conditions and is error prone in particular if it is done manually. Even a quadratic number of such operations might be too large. Separate ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proc. of 11th Conference on Computational Complexity, pages 290--299. IEEE Computer Society Press, 1996.

....using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3 SAT, independent set, and 3 colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 17] The models we define were inspired as well by the work of [3, 23]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multi set of molecular sequences. We describe the allowable operations below. Where set notation is applied to multi sets, ....

....important complexity measure is the solution space size (also called simply space) i.e. the maximumnumber of strings in all test tubes at any time, counting multiplicities. Adleman [2] has speculated that molecular computation with a solution space of size 2 70 might be possible. Recent papers [3, 19] attempt to optimize solution space size for particular combinatorial problems. Problem instances are associated with a parameter n called their size. In complexity theory, n is the length of a suitable encoding of the instance. However, in analysis of algorithms, n is usually a more natural ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....assignment that makes the formula true. The number of molecular strands used by a molecular algorithm is an important measure of the algorithm s complexity. This measure is also called the volume used by the algorithm. The number of operations used in a molecular computation is called time. In [3, 15, 7, 14], efforts were made in optimize the volume of DNA solution. Various papers [19, 6, 17, 8] apply molecular computing to solve standard computational problems. Beaver [6] and Roos and Wagner [18] designed DNA algorithms to solve a PSPACEcomplete problem, but they used an operation that requires ....

....and O(2 n) volume. 2. DNA Computation Model Laboratory techniques for recombinant DNA and RNA manipulation are becoming highly standardized. Basic principles about recombinant DNA can be found in [21, 22] The computational models we define in this section were inspired by the work of [3, 18]. A test tube is a multiset of binary strings. We describe the allowable operations formally below. Where set notation is applied to multisets, multiplicities are respected. In the definitions, T 1 , T 2 , and T 3 denote distinct test tubes and c denotes a character. The running time for a ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3 SAT, independent set, and 3 colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 20] The models we define were inspired as well by the work of [3, 28]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations below. Where set notation is applied to multisets, ....

....the size of the solution space (also called simply volume) which is the maximum number of strings in all test tubes at any time, counting multiplicities. Adleman [2] has speculated that molecular computation with a solution space of size 2 70 (about 0:002 moles) might be possible. Recent papers [3, 23] attempt to optimize solution volume size for particular combinatorial problems. Problem instances are associated with a parameter n called their size. In complexity theory, n is the length of a suitable encoding of the instance. However, in analysis of algorithms, n is usually a more natural ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....with classical problems or not. For example, for the MAX Clique problem, there are algorithms achieving running time of about 2 3 [20, 17] and these will be preferable to the approach presented here which needs 2 strands of DNA. The methods presented in this paper can be combined with results of [4] to produce more efficient DNA algorithms for solving the MAXClique problem. In general, DNA algorithms work for any problem, and hence may be favorable for problems where no algorithms significantly faster than 2 are known. An interesting example of an application where the use of DNA may be ....

....over all non edges e = u, v) of G and throwing away those sets that contain both u and v. At the end of the process we separate the DNA strands by length. The longest DNA strand corresponds to a maximal clique in G. This process requires only n 2 IEI biological steps. We note that recently [4] showed that these techniques combined with a more clever combinatorial algorithm can be used to solve MAX Clique and 3 Coloring more efficiently, i.e. using less than 2 strands for a graph with n vertices. 4.3 Regular Circuit Satisfiability: using state automata As was explained in the ....

E. Bach, A. Condon, E. Glaser and C. Tanguay. DNA Models and Algorithms for NP-Complete Problems. Proceedings of the 11th Annual IEEE Conference on Computational Complexity, 1996.

....52895. Address: Dept. of Electrical Engineering and Computer Science, 19 Memorial Dr W Ste 2, Bethlehem PA 18015 3084. Email: fbeigel bif3g eecs.lehigh.edu. Web: http: www.eecs.lehigh.edu fbeigel bif3g. called the volume (or the size of the solution space) used by the algorithm. Recent papers [3, 16, 4] have attempted to optimize solution space size for particular combinatorial problems. In this paper we begin a systematic study of volume bounded molecular computation, and we determine the relations among several models of molecular computation and Turing machine computation. The molecular ....

....and not volume complexity for molecular computation. Furthermore, only brute force algorithms have been implemented in the memoryless models he describes. 2. Molecular Operations Molecular computation was first studied in [1, 15] The models we define were inspired as well by the work of [3, 19]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations below. Where set notation is applied to multisets, ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....points out in [42] if Adleman s experiment were scaled up to 200 vertices the weight of DNA required would exceed that of the Earth. Mac D onaill also presents an analysis of the scalability of DNA computations in [23] as do Linial and Linial [51] Lo et al. 54] and Bunow [20] We note that [9] have described DNA algorithms which reduce the problem just outlined, however, the exponential curse is inherent in the NP complete problems. There is the hope, as yet unrealised (despite the claims of [11] that for problems in the complexity class P (i.e. those which can be solved in ....

Eric Bach, Anne Condon, Elton Glaser, and Celena Tanguay. DNA Models and Algorithms for NP-complete Problems, pages 290--299. IEEE Computer Society Press, 1996.

....of n di erent coupons by randomly drawing a coupon each trial. You can expect to make O(n log n) drawings before you collect the entire set. Similarly, the amount of DNA required to create 19 all 2 n potential solutions is actually O(n2 n ) This was rst noted in Linial and Linial (1995) Bach et al. 1996) show that Lipton s (1995) algorithm for generating random solutions reduces to the coupon collectors problem. Even though there is some bias in the process due to the equal amounts of 0 and 1 subsequences added to the test tube, the rst 2 n 1 solutions form in a relatively unbiased fashion. ....

Bach, E., Condon, A., Glaser, E., and Tanguay, C. (1996). DNA Models and Algorithms for NP-complete Problems, pages 290-299. IEEE Computer Society Press, New York, NY.

....in order for DNA based computation to compete against silicon based computers. Several proposals have been made in the past (see [ORS97,OR99] for a survey) to resolve the problem of reducing the amount of DNA to be used. In particular, the following three proposals seem important: 1) Bach et al. BCGT96] proposed methods for solving such NP complete problems as Clique and Independent Set, where solution candidates to the instance is generated by combining (in all possible combinations) solutions for its subproblems. 2) Ogihara [Ogi96] see also [OR97] proposed breadth rst search methods for ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of 11th Conference on Computational Complexity, pages 290-299. IEEE Computer Society Press, Los Alamitos, CA, 1996.

....length. Indeed we can not expect to reduce the solution space for such problems to polynomial size with optimized algorithms because this would imply the equality of P and NP [43] nevertheless the number of the actually computable instances grows considerably. Bach, Condon, Glaser, and Tanguay [5] present clever algorithms that use Adleman s operations and test. For example they reduce the DNA consumption to an order of 1:51 n while computing the NP complete Independent Set problem. Ogihara utilizes the Monien Speckenmeyer algorithm to construct a DNA algorithm for 3 SAT that increases ....

....PCR to a series of test tubes instead of only one. Boneh and Lipton [13] make their 3 SAT algorithm error resistant by duplicating the number of strands periodically. Among the first to be engaged in the error analysis of Adleman s and Lipton s operations are Bach, Condon, Glaser, and Tanguay [5]. They increase the probability so when duplicating test tubes indeed all different strands are in both of the tubes, by using a greater amount of equal molecules. In [7] Baum is concerned with the coding of strings in DNA strands. Using strands of sufficent length it can happen that randomly ....

E. Bach, A. E. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proc. of 11th Conference on Computational Complexity, pages 290--299, 1996.

.... Journal of Chemical Physics, 809, 1156, 312, 69, 241] Journal of Chemical Technology and Biotechnology, 58] Journal of Chemometrics, 299] Journal of Computational Chemistry, 37, 40, 48, 49, 51, 968, 994, 65, 1007, 1121, 1166, 68, 81, 101, 102, 1109] Journal of Computer and System Sciences, [610] Journal of Computer Aided Molecular Design, 271, 272, 965, 1150, 1153, 1022, 1174, 197] Journal of General Virology, 435] Journal of Geomagnetism and Geoelectricity, 560] Journal of Global Optimization, 412] Journal of Japanese Society for Arti cial Intelligence, 625] Journal of ....

....[643, 684] Aspn as, Anders, 981] Ator, Mark A. 296] Aunger, P. 302] Autere, Antti, 265] Axmann, Joachim K. 754, 761, 56, 768, 769, 771, 774] Aye, T. M. 854] Ayers, L. 24] Aygun, K. 498] Azzaro Pantel, Catherine, 83] Azzaropantel, C. 84] Babenko, Vladimir, 700] Bach, Eric, [610] B ack, Thomas, 657, 765] Backofen, R. 595] Backofen, Rolf, 595] Baleja, James D. 115, 328] Balogh, S. 309] Banerjee, S. 682] Bangalore, Arjun S. 1113] Bangalore, Shanthamallikarjuna Shivappa, 1118] Bansal, A. 596] Banzhaf, Wolfgang, 887, 888, 640, 641] B ar, R. 97] ....

[Article contains additional citation context not shown here]

Eric Bach, Anne Condon, Elton Glaser, and Celena Tanguay. DNA models and algorithms for NP-complete problems. Journal of Computer and System Sciences, 57(2):172-186, October 1998. ga98aEricBach.

....methods. One approach to reduce this cost is to build duplicate sub sequences only once. Sophisticated ways of eciently creating large sets of oligonucleotides also exist. For example, methods have been proposed to eciently synthesize many oligos for the purposes of molecular computation, e.g. [Bach et al. 1996]. This introduces a notion of cost to the output of our algorithm outputs. That is, certain legitimate sets of Sdna i and Ldna i may be cheaper to synthesize than others. 8 Acknowledgements We would like to thank Steve Rozen for his many helpful comments. Michael Benedik and Mike MacDonell have ....

Bach, E., Condon, A., Glaser, E. and Tanguay, C. 1996. DNA Models and Algorithms for NP-complete Problems, 290-299. In IEEE, 1996 IEEE Conference on Computational Complexity. IEEE Computer Society Press, Los Alamitos, USA.

.... be that a hybrid approach will really be necessary to give the major benefits of each; Reif suggests such an approach, combining self assembly steps with manual laboratory steps [36] Combinations of one or more molecular computing methods with electronic computation, as was done by Bach et al. [7], seem even more promising. None of the methods considered seems likely to be practical without much more experimental work and improvements in methodology, but any combination of them might ultimately overcome their practical difficulties. We can also note that we have considered only three ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity, pages 290--299, 1996.

....into a simpler one under the sticker based model. Another promising application is dynamic programming. With the method presented in [7] an n item Knapsack instance with the total weight sum W and the total value sum V can be solved in time O(n) with the volume O(V W ) of DNA. The paper [6] presents a 3 Coloring algorithm, which uses the total volume n1:89 n of DNA and assumes an operation for splitting test tube contents into weighted subsets. The paper also presents a volume 1:67 n algorithm and a volume 1:51 n algorithm for Independent Set. The paper [30] shows that a ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proc. 11th Conference on Comp. Complexity, pages 290--299. IEEE Computer Society Press, 1996.

....assignment that makes the formula true. The number of molecular strands used by a molecular algorithm is an important measure of the algorithm s complexity. This measure is also called the volume used by the algorithm. The number of operations used in a molecular computation is called time. In (Bach et al. 1996, Ogihara 1996, Beigel and Fu 1997, Morimoto 1996) efforts were made in optimize the volume of DNA solution. Various papers (Roweis et al. 1996, Beaver 1995, Reif 1995, Boneh et al. 1995) apply molecular computing to solve standard computational problems. Beaver (Beaver 1995a) and Roos and Wagner ....

....2. DNA Computation Model Laboratory techniques for recombinant DNA and RNA manipulation are becoming highly standardized. Basic principles about recombinant DNA can be found in (Watson et al. 1987, Watson et al. 1992) The computational models we define in this section were inspired by the work of (Bach et al. 1996) A test tube is a multiset of binary strings. We describe the allowable operations formally below. Where set notation is applied to multisets, multiplicities are respected. In the definitions, T 1 , T 2 , and T 3 denote distinct test tubes and c denotes a character. The running time for a ....

[Article contains additional citation context not shown here]

Bach, E., Condon, A., Glaser, E., and Tanguay C., 1996, DNA models and algorithms for NP-complete problems. In proceedings of the 11th annual conference on structure in complexity theory, 290-299.

....classical problems or not. For example, for the MAX Clique problem, there are algorithms achieving running time of about 2 n=3 [20, 17] and these will be preferable to the approach presented here which needs 2 n strands of DNA. The methods presented in this paper can be combined with results of [4] to produce more efficient DNA algorithms for solving the MAXClique problem. In general, DNA algorithms work for any problem, and hence may be favorable for problems where no algorithms significantly faster than 2 n are known. An interesting example of an application where the use of DNA may be ....

....over all non edges e = u; v) of G and throwing away those sets that contain both u and v. At the end of the process we separate the DNA strands by length. The longest DNA strand corresponds to a maximal clique in G. This process requires only n 2 0 jEj biological steps. We note that recently [4] showed that these techniques combined with a more clever combinatorial algorithm can be used to solve MAX Clique and 3 Coloring more efficiently, i.e. using less than 2 n strands for a graph with n vertices. 4.3 Regular Circuit Satisfiability: using state automata As was explained in the ....

E. Bach, A. Condon, E. Glaser and C. Tanguay. DNA Models and Algorithms for NP-Complete Problems. Proceedings of the 11th Annual IEEE Conference on Computational Complexity, 1996.

....molecular computation. 1. Introduction Molecular computation was first studied in [1, 15] which identified the number of molecular strands used as an important resource. This measure is also called the space (or the size of the solution space) used by the algorithm. Recent papers [3, 16, 4] have attempted to optimize solution space size for particular combinatorial problems. In this paper we begin a systematic study of space bounded molecular computation, and we determine the relations among several models of molecular computation and Turing machine computation. The molecular ....

....and not space complexity for molecular computation. Furthermore, only brute force algorithms have been implemented in the memoryless models he describes. 2. Molecular Operations Molecular computation was first studied in [1, 15] The models we define were inspired as well by the work of [3, 19]. A molecular sequence is a string over an alphabet S (we can use any alphabet we like, encoding characters of S by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations below. Where set notation is applied to multisets, ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay.DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3 SAT, independent set, and 3 colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 18] The models we define were inspired as well by the work of [3, 24]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multi set of molecular sequences. We describe the allowable operations below. Where set notation is applied to multi sets, ....

....measure is the solution space size (also called simply space) i.e. the maximum number of strings in all test tubes at any time, counting multiplicities. Adleman [2] has speculated that molecular computation with a solution space of size 2 70 (about 0:002 moles) might be possible. Recent papers [3, 20] attempt to optimize solution space size for particular combinatorial problems. Problem instances are associated with a parameter n called their size. In complexity theory, n is the length of a suitable encoding of the instance. However, in analysis of algorithms, n is usually a more natural ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proc. 11th Ann. Conf. Structure in Complexity Theory, pp. 290--299, 1996.

....molecular computation. 1. Introduction Molecular computation was first studied in [1, 15] which identified the number of molecular strands used as an important resource. This measure is also called the space (or the size of the solution space) used by the algorithm. Recent papers [3, 16, 4] have attempted to optimize solution space size for particular combinatorial problems. In this paper we begin a systematic study of space bounded molecular computation, and we determine the relations among several models of molecular computation and Turing machine computation. The molecular ....

....and not space complexity for molecular computation. Furthermore, only brute force algorithms have been implemented in the memoryless models he describes. 2. Molecular Operations Molecular computation was first studied in [1, 15] The models we define were inspired as well by the work of [3, 19]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations below. Where set notation is applied to multisets, ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3 SAT, independent set, and 3 colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 20] The models we define were inspired as well by the work of [3, 28]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations below. Where set notation is applied to multisets, ....

....the size of the solution space (also called simply volume) which is the maximum number of strings in all test tubes at any time, counting multiplicities. Adleman [2] has speculated that molecular computation with a solution space of size 2 70 (about 0:002 moles) might be possible. Recent papers [3, 23] attempt to optimize solution volume size for particular combinatorial problems. Problem instances are associated with a parameter n called their size. In complexity theory, n is the length of a suitable encoding of the instance. However, in analysis of algorithms, n is usually a more natural ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....one under the sticker based model (see Section 4) Another promising application is dynamic programming. With the method presented in [BB] an n item Knapsack instance with the total weight sum W and the total value sum V can be solved in time O(n) with the volume O(V W ) of DNA. The paper [BCGT96] presents a 3 Coloring algorithm, which uses the total volume n1:89 n of DNA and assumes an operation for splitting test tube contents into weighted subsets. The paper also presents a volume 1:67 n algorithm and a volume 1:51 n algorithm for Independent Set. The paper [Ogi96] shows that a ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NPcomplete problems. In Proceedings of 11th Conference on Computational Complexity, pages 290--299. IEEE Computer Society Press, 1996.

....expect to reduce the solution space for such problems to polynomial size even with optimal algorithms because this would imply the equality of P and NP [49] Nevertheless, better algorithms can raise the number of the actually computable instances considerably. Bach, Condon, Glaser, and Tanguay [4] present clever algorithms that use Adleman s operations and test. For example they reduce the DNA consumption from an order of 3 n (following Adleman in [2] to 2 n while computing the NP complete 3 Coloring problem. A graph is called to be 3 colorable if its vertices can be painted in three ....

....PCR to a series of test tubes instead of only one. Boneh and Lipton [12] make their 3 SAT algorithm error resistant by duplicating the number of strands periodically. Among the first to be engaged in the error analysis of Adleman s and Lipton s operations are Bach, Condon, Glaser, and Tanguay [4]. They present a probabilistic model for the duplication of test tubes. Indeed duplicating is a process of splitting test tubes. Lipton assumes each amount of each strand in an old tube to be divided equally to the new tubes. In the prob18 abilistic model each possible partition of the old test ....

E. Bach, A. E. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Conference on Computational Complexity, pages 290--299, 1996.

....assignment that makes the formula true. The number of molecular strands used by a molecular algorithm is an important measure of the algorithm s complexity. This measure is also called the volume used by the algorithm. The number of operations used in a molecular computation is called time. In [3, 15, 7, 14], efforts were made in optimize the volume of DNA solution. Various papers [19, 6, 17, 8] apply molecular computing to solve standard computational problems. Beaver [6] and Roos and Wagner [18] designed DNA algorithms to solve a PSPACEcomplete problem, but they used an operation that requires ....

....n n 2 log 2 n) volume. 2. DNA Computation Model Laboratory techniques for recombinant DNA and RNA manipulation are becoming highly standardized. Basic principles about recombinant DNA can be found in [21, 22] The computational models we define in this section were inspired by the work of [3, 18]. A test tube is a multiset of binary strings. We describe the allowable operations formally below. Where set notation is applied to multisets, multiplicities are respected. In the definitions, T 1 , T 2 , and T 3 denote distinct test tubes and c denotes a character. The running time for a ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....52895. Address: Dept. of Electrical Engineering and Computer Science, 19 Memorial Dr W Ste 2, Bethlehem PA 18015 3084. Email: fbeigel bif3g eecs.lehigh.edu. Web: http: www.eecs.lehigh.edu fbeigel bif3g. called the volume (or the size of the solution space) used by the algorithm. Recent papers [3, 16, 4] have attempted to optimize solution space size for particular combinatorial problems. In this paper we begin a systematic study of volume bounded molecular computation, and we determine the relations among several models of molecular computation and Turing machine computation. The molecular ....

....and not volume complexity for molecular computation. Furthermore, only brute force algorithms have been implemented in the memoryless models he describes. 2. Molecular Operations Molecular computation was first studied in [1, 15] The models we define were inspired as well by the work of [3, 19]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations below. Where set notation is applied to multisets, ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....were introduced by Adleman [1, 8] but so far the field lacks a killer application. It is well known that a DNA computer can solve SAT in linear time [8] but using an exponential number of DNA strands. The number of strands used by an algorithm is called the volume. Although recent papers [5, 4, 9] solve NP problems using smaller exponential volume, we believe that it is essential to find applications of DNA computers that use subexponential volume. In this paper we apply DNA computers to the approximate solution of NPoptimization problem. We show how DNA algorithms using subexponential ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....were introduced by Adleman [1, 8] but so far the field lacks a killer application. It is well known that a DNA computer can solve SAT in linear time [8] but using an exponential number of DNA strands. The number of strands used by an algorithm is called the volume. Although recent papers [5, 4, 9] solve NP problems using smaller exponential volume, we believe that it is essential to find applications of DNA computers that use subexponential volume. In this paper we apply DNA computers to the approximate solution of NP optimization problem. We show how DNA algorithms using subexponential ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....into a simpler one under the sticker based model. Another promising application is dynamic programming. With the method presented in [BB] an n item Knapsack instance with the total weight sum W and the total value sum V can be solved in time O(n) with the volume O(V W ) of DNA. The paper [BCGT96] presents a 3 Coloring algorithm, which uses the total volume n1:89 n of DNA and assumes an operation for splitting test tube contents into weighted subsets. The paper also presents a volume 1:67 n algorithm and a volume 1:51 n algorithm for Independent Set. The paper [Ogi96] shows that a ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NPcomplete problems. In Proceedings of 11th Conference on Computational Complexity, pages 290--299. IEEE Computer Society Press, 1996.

....the complexity is 2 n . So, both Adleman s method and Lipton s method can handle inputs of size up to 60. However, such small instances are in most cases manageable by electronic computers. Efforts have been made to reduce the amount of volume for DNA computers to solve NP complete problems [BCGT96,Ogi96,BF96] In this paper, we focus our attention to 3SAT. The current fastest sequential algorithm for 3SAT runs in 1:5 n steps [Kul96] If the algorithm is implemented on a computer that executes a million instructions in a second, the formulas of 60 variables will require approximately 10 ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Conference on Computational Complexity, pages 290--299. IEEE Computer Society Press, 1996.

....1. Introduction Adleman s pioneering experiment [1] opened the possibility that moderately large instances of NP complete problems might be solved via techniques from molecular biology. Since then numerous papers have explored more efficient molecular algorithms for particular problems in NP [27, 10, 3, 30, 8, 20, 21, 18], molecular solutions to PSPACE complete problems [7, 37] and fault tolerant molecular algorithms [12, 25] Other papers have examined the relationships between molecular complexity classes and classical complexity classes [38, 19] We will survey some of these advances in this paper. For ....

....all strands not containing S v (see Section 3.4) 5. Amplify the solution by the polymerase chain reaction (see Section 3.5) Output yes if the solution contains at least one DNA sequence; no, otherwise. 5. Molecular Computation Models The models we define were inspired by the work of [3, 38]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multiset of molecular sequences. We describe the allowable operations formally below. Where set notation is applied to ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....problem Q, one has only to transform 3SAT instances to those for Q and apply the DNA algorithm to obtain an answer. Unfortunately, not many algorithms for NPcomplete problems have been developed so far. Only known algorithms are those for Maximum Independent Set (MIS) and 3 Coloring by Bach et al. [BCGT96], with space complexity 2 cn for some c 1. However, the space complexity of these algorithms is not small enough to transform 3SAT. The standard transformations of 3SAT to these problems map a formula of n variables and m clauses to an instance of size 2n 3m. Since m can be as large as 8 ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Conference on Computational Complexity, pages 290--299. IEEE Computer Society Press, 1996.

....on test tubes. The volume is the maximum number of strings in all test tubes at any time, counting multiplicities. The strand length complexity of a molecular algorithm is the length of the longest DNA strand used in the computation. Although time and volume complexity have been well studied [13, 6, 2, 14, 5, 9, 10, 8], strand length has received less attention. Yet Roweis et al. [16] state that 2500 base sequences decay at a rate of 10 per hour, and Sambrook [17] states that DNA strands longer than 10000 bases have a serious decay problem. In their survey, Ogihara et al. [15] list strand length complexity as ....

....recurrence can be solved explicitly to obtain t(n) 1:47 n . We implement Tarjan s algorithms for independent set problem and have the following theorem. We decrease the length complexity to O(n) and keep the same time and volume complexity. This improves previous algorithms for this problem [2, 5]. Theorem 14. There is a fSeparate; Merge; Append; Cut; Initg molecular algorithm for Independent set problem with O(jGj) length, O(n G ) time and 1:47 n volume. Implementing Monien and Speckenmeyer s algorithm for 3SAT problem, we have the following theorem. The length used in their ....

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3 SAT, independent set, and 3 colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 17] The models we define were inspired as well by the work of [3, 23]. A molecular sequence is a string over an alphabet Sigma (we can use any alphabet we like, encoding characters of Sigma by finite sequences of base pairs) A test tube is a multi set of molecular sequences. We describe the allowable operations below. Where set notation is applied to multi sets, ....

....complexity measure is the solution space size (also called simply space) i.e. the maximumnumber of strings in all test tubes at any time, counting multiplicities. Adleman [2] has speculated that molecular computation with a solution space of size 2 70 might be possible. Recent papers [3, 19] attempt to optimize solution space size for particular combinatorial problems. Problem instances are associated with a parameter n called their size. In complexity theory, n is the length of a suitable encoding of the instance. However, in analysis of algorithms, n is usually a more natural ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

....can only deal with inputs of small size. Hartmanis [27] showed that a mass of DNA greater than that of the earth would be required to solve a 200 vertex instance of the Hamiltonian path problem. Solving NP complete problems with molecular computation was further studied in a series of papers [13, 3, 39, 9, 48, 38, 40]. In [3, 39, 9, 38] efforts were made in optimize the volume of DNA solution. Applying a molecular computer to solving various computational problems was studied in a series of papers [48, 5, 44, 12] In [12] a molecular algorithm was designed for breaking the Data Encryption Standard (DES) ....

....of small size. Hartmanis [27] showed that a mass of DNA greater than that of the earth would be required to solve a 200 vertex instance of the Hamiltonian path problem. Solving NP complete problems with molecular computation was further studied in a series of papers [13, 3, 39, 9, 48, 38, 40] In [3, 39, 9, 38], efforts were made in optimize the volume of DNA solution. Applying a molecular computer to solving various computational problems was studied in a series of papers [48, 5, 44, 12] In [12] a molecular algorithm was designed for breaking the Data Encryption Standard (DES) Breaking DES means ....

[Article contains additional citation context not shown here]

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, pages 290--299, 1996.

No context found.

Eric Bach, Anne Condon, Elton Glaser, and Celena Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings, Eleventh Annual IEEE Conference on Computational Complexity, pages 290--300, Philadelphia, Pennsylvania, 24--27 May 1996. IEEE Computer Society Press.

No context found.

E. Bach, A. Condon, E. Glaser, and C. Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity, pages 290-300, 1996.

No context found.

E. Bach, A. Condon, E. Glaser, C. Tanguay, DNA models and algorithms for NP-complete problems, in: Proceedings, Eleventh Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Philadelphia, Pennsylvania, 1996, pp. 290--300.

No context found.

Eric Bach, Anne Condon, Elton Glaser, and Celena Tanguay. DNA models and algorithms for NP-complete problems. In Proceedings of the 11th Conference on Computational Complexity, pages 290--299. IEEE Computer Society Press, 1996.

No context found.

Eric Bach, Anne Condon, Elton Glaser, and Celena Tanguay. DNA Models and Algorithms for NP-complete Problems, pages 290--299. IEEE Computer Society Press, 1996.

No context found.

E. Bach, A. Condon, E. Glaser, C. Tanguay. DNA Models and Algorithms for NP-complete Problems, IEEE Computer Society Press, pages 290 -- 299, 1996.

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