Testing Whether a Set of Code Words Satisfies a Given Set of Constraints *

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1 JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 6, (010) Testing Whether a Set of Code Words Satisfies a Given Set of Constraints * HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO + Department of Computer Science Nationa Tsing Hua University Hsinchu 300, Taiwan E-mai: {bertha, wanchen, peggy, wshih}@rtab.cs.nthu.edu.tw + Department of Eectrica Engineering and Computer Science Northwestern University Evanston 6008, U.S.A. E-mai: ao@northwestern.edu This paper investigates the probem of testing whether a set of code words satisfies certain bioogicay motivated Hamming distance constraints. The paper provides three efficient techniques to verify the code words, namey, the Enumeration, Tabe Looup, and Encoding methods, with appications to the design of DNA words. The Enumeration method enumerates a combinations of positions in a word, so that a the words in a set can be compared simutaneousy and the testing process is improved. The Tabe Looup method constructs a data tabe and divide each word into sub-words to reduce the time compexity of the testing process. The Encoding method which is simiar to Tabe Looup method uses a ined ist to store necessary information in addition. The proposed methods run in O(n) ~ O(og og n) times faster than the naive method when = O(og n), where n is the number of code words in a set and is the ength of a word. Keywords: DNA verification, DNA word design, code word verification, distance constraint, testing agorithm 1. INTRODUCTION The scaabe design of code words has been investigated by many researchers. Huffman [6] presented the we-nown two-pass agorithm to encode variabe-ength source codes. A new one-pass agorithm for constructing dynamic Huffman codes is then proposed in [13]. Yang et a. [14] introduce an efficient code word design method for the shortest common superstring probem which is proved to NP-hard [4]. In recent years, many code word design methods are appied to sove DNA-reated probems [5, 9-11]. For instance, one is for detecting protein-dna-binding sites [9, 10]. A number of different proteins bind into a genome. Some of these DNA-binding proteins ony bind to specific ocations where they have to execute certain functions. Another is for designing error-preventing DNA sequence [5, 11]. The potentia errors in DNA sequence shoud be minimized for reiabe DNA computing. Received March 31, 008; revised May 9, 008; accepted May 15, 008. Communicated by Tsan-sheng Hsu. * A part of the paper has been presented in Nationa Computer Symposium, NCS Dec. 007, Asia University, Taichung, Taiwan. The conference is sponsored by Asia University, Nationa Science Counci, Institute of Information Science of Academia Sinica,..., etc. 333

2 334 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO In this paper, we study the verification agorithm for designing DNA words. For the DNA word design probem, many agorithms have been deveoped. Marathe et a. [8] used a dynamic programming approach to design DNA sequences based on Hamming distance and free energy. Deaton et a. [3] used genetic search to find good encodings. Arita et a. [1] deveoped a DNA sequence design system using genetic agorithms and random generate-and-test agorithms. Tanaa et a. [1] isted sequence fitness criteria and sequences are generated using simuated anneaing method. Recenty, Kao et a. [7] considered more fitness criteria for DNA sequence design. Since some DNA word design agorithms are randomized, we need a fast verification agorithm to verify whether the output indeed satisfy the given set of constraints. Therefore, we focus on the deveopment of a fast verification agorithm that can determine whether a set of words satisfy a given set of constraints. In this paper, five constraints which are denoted as C 1, C, C 3, C 4, and C 5 are considered and these constraints can be cassified as basic Hamming distance constraint, reverse compement Hamming distance constraint, sef-compementary Hamming distance constraint, and shifting Hamming distance constraint which are proposed in [, 7, 8] for DNA word designs. To verify these constraints for a set of DNA words designed, we propose the three foowing techniques: (1) Enumeration. It enumerates a possibe combinations of + 1 positions in a word, where is the ength of a word and is the parameter for determining whether any two words in the set meet the given constraint. Note that, since the ength of a word is, then the vaue of shoud greater than zero and ess than, otherwise, the verification for DNA words is meaningess. This method determines whether the set of words satisfies the given constraints by checing whether any two sub-words seected from a possibe combinations of + 1 positions in the two words of the set are the same. () Tabe Looup. It first enumerates a possibe sub-words with a fixed ength over a given aphabet and then computes the Hamming distance between any two sub-words in the enumeration for constructing a data tabe. By dividing each word into some sub-words and ooing up the constructed data tabe, the Hamming distance between two words can be obtained. (3) Encoding. It utiizes a tabe and a ined ist for storing some necessary information. It first divides each word into some sub-words with a fixed ength and puts them into different groups. If the sub-words of different words start at the same position in the words, then the sub-words are put into the same group. Each group of sub-words are assigned a unique group ID. By sorting the sub-words in a group, it determines whether any two sub-words are the same. If a group contains two different sub-words, then update the corresponding vaue in the tabe and store the group ID into the corresponding ocation of the ined ist. After that, this method determines whether a set of words satisfies the given constraints by checing the tabe and ined ist. The time compexities of these methods for given constraints are summarized as Tabe 1. Tabe 1. Time compexities of each method for different constraints. Enumeration Tabe Looup Encoding C 1, C min{ 1, 1} On ( *( )* + ) * O( * + n * ) C 3, C min{ 1, 1} 4 On ( *( )* + * ) O( * * + n* * ) C 5 N.A. n** og On ( * * ) On ( * * * ) O ( *og + ) N.A.

3 TESTING CODE WORDS WITH CONSTRAINTS 335 The rest of this paper is organized as foows: Section, provides basic definitions of constraints we dea with. In section 3, we propose Enumeration method to verify the given set of code words under the given constraints. Tabe Looup method and the Encoding method are presented in sections 4 and 5, respectivey. The paper is concuded in section 6.. PRELIMINARIES In this paper we dea a set of words W with two aphabets, namey, the binary aphabet {0, 1} and the DNA aphabet {A, C, G, T}. A given set of words W is assumed to each has ength and et W = n. Let X = x 1 x x denote a word, where x i denotes the ith position in X and is aso caed a character of X. X denotes the ength of X, i.e., X =. Let X[i..j] = x i x i+1 x j denote a sub-word of X, 1 i j. If i = j, then X[i..i] = X[i] = x i. The reverse of X, denoted by X R, is the word x x -1 x 1, and the sub-word of the reverse of X is denoted by X R [i..j], e.g., X R [1..3] = x x -1 x -. The compement of X, denoted by X C, is the word x 1 C x C x x C, and the sub-word of compement of X is denoted by X C [i..j], where x i C is defined as 0 C = 1 and 1 C = 0, if x i is over the binary aphabet; A C = T, C C = G, G C = C, T C = A, if x i is over the DNA aphabet. The reverse compement of X is the compement of X R, X RC = x C x -1 C x 1 C, and the sub-word of compement of X R is denoted by X RC [i..j], e.g., X RC [1..] = x C x -1 C. Note that, X[i..j] RC, denotes the reverse compement of the sub-word of X, e.g., X[1..] RC = x C x 1 C. Let + be the operation that concatenates any two words X = x 1 x x and Z = z 1 z z i, such that X + Z = x 1 x x z 1 z z i, where i > 0, and X + Z = + i. The Hamming distance between two words X and Y = y 1 y y, which is defined as d(x, Y) = {i x i y i, 1 i }, is the number of positions where X differs from Y. We are interested in verifying a given set of words W satisfies the foowing constraints [7]. Note that the foowing constraints can prevent unwanted hybridization between two DNA standards and each of them is denoted as a function C f ( f ) with a given parameter f : 1. Basic Hamming Distance Constraint ( 1 ): C 1 ( 1 ) = {d(y, X) 1 X, Y W, X Y}.. Reverse Compementary Constraint ( ): C ( ) = {d(y, X RC ) X, Y W, X Y}. 3. Shifting Hamming Constraint ( 3 ): C 3 ( 3 ) = {d(y[1..i], X[( i + 1).. ]) 3 ( i) 1 i, X, Y W, X Y}. 4. Shifting Reverse Compementary Constraint ( 4 ): C 4 ( 4 ) = {d(y[1..i], X[1..i] RC ) 4 ( i); and d(y[( i + 1).. ], X[( i + 1).. ] RC ) 4 ( i) 1 i, X, Y W, X Y}. 5. Shifting Sef Compementary Constraint ( 5 ): C 5 ( 5 ) = {d(y[1..i], Y[1..i] RC ) 5 ( i); and d(y[( i + 1).. ], Y[( i + 1).. ] RC ) 5 ( i) 1 i, Y W}. Note a naive agorithm for testing constraints C 1 and C taes O(n ) time; for testing constraints C 3 and C 4 taes O(n * * ) time; and for testing C 5 taes O(n * * ) time, where is the parameter of the given constraint. In this paper, we propose three techniques for soving the verification of a given set of words. The proposed approaches determine whether a set of words satisfies a set of combinatoria constraints faster than the naive agorithms on severa practica cases such as = O(og n). The first is Enu-

4 336 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO meration method which can be appied to constraints C 1 C 4. Whie = O(og n) and the og parameters of given constraints are no greater than ( n O og og n ), the time compexity of the Enumeration method is ower than a naive agorithm and the other methods which proposed in here. The second is Tabe Looup method which can be appied to constraints C 1 C 5. The Tabe Looup method is an efficient approach whie = O(og n) and its time compexity is ower than that of a naive agorithm for any constraint if is no greater than O(og n). Finay, the Encoding method can be appied to C 1 C 4 and its time compexity is ower than that of naive agorithm whie the order of is ess than. This method is better than the other methods whie = Ω(og n). The comparison of time compexities between naive agorithms and our techniques is summarized in Tabe. Note that, it is reasonabe to assume = O(og n), since it has been shown that a set of n words of ength O(og n + ) can be generated to fufi the constraints with high probabiity [7]. Tabe. The comparison of time compexities between naive agorithms and our agorithms assuming = O(og n) and ε is any constant ess than1. #1: Using Enumeration method, #: Using Tabe Looup method. og O(og n) ( n O ) O(og og n) O(1) C 1, C C 3, C 4 C 5 og og n naive O(n * og n) O(n * og n) O(n * og n) O(n * og n) ours On ( * og n );# O(n ); #1 O(n 1+ε ); #1 O(n * og n); #1 naive O(n * og n) ours O(n * og n); # naive O(n * og n) ours n*og n og ogn O ( );# og n og og n og n og og n On ( * ) O(n * og n * og og n) O(n * og n) On ( * );#1 O(n 1+ε * og og n); #1 O(n * og n); #1 og n og og n n*og n (og og n) On ( * ) O(n * og n * og og n) O(n * og n) O( );# O(n * og n); # n*og n og og n O ( );# 3. ENUMERATION First, our discussion is based on C 1 constraint and using the binary aphabet. For any two words X and Y in W, our probem is to test whether d(x, Y), where is the given parameter of the corresponding constraint. Our basic idea is that if d(x, Y) <, it means there are at east + 1 positions that the same in X and Y. Hence, we seect any two sub-words with ength + 1 from X and Y respectivey, and then chec if they are the same. If any two sub-words with ength + 1 of X and Y are not the same, then d(x, Y) ; otherwise, d(x, Y) <. There are totay ( + 1) combinations to seect a sub-word with ength + 1 from a word with ength. Without oss of generaity, we assume that any sub-word with ength + 1 of a word with ength is corresponding to a unique combination of + 1 positions in a word. Each combination of + 1 positions in a word is assigned a unique abe p, which is caed pth combination, where p = 1,,, ( + 1), e.g., 1st combination is 1st, nd,, ( + 1)th positions in a word and nd combination is 1st, 3rd,, ( + )th positions in a word. The detai of the Enumeration

5 TESTING CODE WORDS WITH CONSTRAINTS 337 Agorithm 1 Enumeration Agorithm 1. procedure Enumeration (W) {*Output Fai, if the constraint is not satisfied; Success, if the constraint is satisfied *}. for p from 1 to ( + 1) do 3. Let W p = {X 1,p, X,p,, X n,p } contains n sub-words with ength + 1 in W which are seected according to the pth combination, where = 1, X i,p is the sub-word of pth combination of X i, i = 1,,, n; 4. Perform radix sort on W p ; 5. if there are any two sub-words X i,p and X j,p, i j in W p are the same in the sorted resut then 6. return Fai ; 7. end if 8. end for 9. return Success ; 10. end procedure method for C 1 constraint is shown in Agorithm 1 and this agorithm can be easiy extended to dea with C constraint. Here is an exampe to show how to verify a set of words W satisfies C 1 ( 1 ) by Agorithm 1. Assume there are three words X 1 = , X = , and X 3 = in W. For C 1 (5), i.e., 1 = 5, there are ( ) = 70 combinations to seect 3 sub-words with 4 characters, + 1 = = 4, from W. When the combination of positions in a word are 1st, 3rd, 6th and 8th positions, the corresponding sub-words of the 3 words in W are 1100, 1100, 0100 (from X 1, X, and X 3, respectivey). After using radix sort at step 4, we find that two sub-words are equa (i.e., 1100), and therefore W does not satisfy constraint C 1 (5). Theorem 1 Agorithm 1 is correct and its time compexity is O(n * ( ) * min{-1, +1} ), where is the given parameter of the corresponding constraint. Proof: For any two words X and Y, the sub-words of X and Y with ength + 1 are compared in steps -8. If there exists two sub-words with ength + 1 that are equa, there must be at most 1 different positions between the two words. Therefore, it does not satisfy the Hamming distance constraint; otherwise, it does. Hence the agorithm is correct. In Agorithm 1, there are ( + 1) combinations to enumerate a possibe combinations of sub-words with ength + 1 in W. Note that, ( + 1) min{-1, +1}. The sub- words of a given pth combination with ength + 1 in W are using radix sort at step 4 in O(n * ( )) time for each p from 1 to ( + 1). Hence, the tota time compexity of Agorithm 1 is O(n * ( ) * min{-1, +1} ). The Enumeration technique can aso be used to perform testing of C 3 constraint. Note that, if the ength i of any two sub-words is ess than, where = 3, then any two sub-words with ength i must satisfy the Hamming distance constraint, since, ( i) < 0. In other words, the agorithm ony needs dea the case that the ength of a sub-word is greater than. For exampe, testing whether a given set of words W satisfies C 3 constraint, the agorithm ony needs to perform testing on any two words X a [1..i] and X b [(

6 338 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO i + 1).. ] with ength i, for i = + 1, +,,, where X a, X b W. Without oss of generaity, et W = {X 1, X,, X n } be a given set of words and W[1..i] = {X 1 [1..i], X 1 [( i + 1).. ], X [1..i], X [( i + 1).. ],, X n [1..i], X n [( i + 1).. ]} be a set of sub-words, for i = + 1,,. It is cear to see that W can be verified by using W[1..i] as an input of Agorithm 1, for i from + 1 to. Therefore, according to Theorem 1, the time compexity of testing whether a set of given words satisfies constraint C 3 is O(n * ( ) * min{-1, +1} * ), where is the given parameter of the constraint. Simiary, Agorithm 1 can be extended to dea with C 4 constraint in O(n * ( ) * min{-1, -+1} * ) time, where = TABLE LOOKUP The method, which determines Hamming distance between any two words by ooing up a precomputed tabe, is caed the Tabe Looup method. There are two data structures needed in this method: (1) Information tabe, which records the Hamming distance between any two words in W, is caed INF tabe. There are n entries in the tabe, et (i, j) denote an entry in row i and coumn j in the INF tabe. Let t i,j denote the vaue of the entry i, j in the INF tabe which is the Hamming distance between the two words X i and X j. Note that, the initia vaue of t i,j is zero, for a 1 i, j n. () Comparison tabe, which enumerates a possibe combinations of sub-words of a given ength, and provides the Hamming distance of any two sub-words in the enumeration, is caed CMP tabe. For exampe, Fig. 1 shows Hamming distance of any two sub-words with ength 3 over the binary aphabet. It is obvious to see that there are 3 sub-words in the enumeration and 3 * 3 entries in the CMP tabe, since the ength of a sub-word is 3 and the size of the binary aphabet is. The first step of Tabe Looup method is to enumerate a possibe sub-words over the given aphabet to construct the CMP tabe. Without oss of generaity, we assume that the given aphabet is binary. Note that each entry in the CMP tabe records the Hamming distance between any two sub-words exhaustivey enumerated. Second, divide each word X i W into sub-words X i [a..b] with fixed ength, where a = (g 1) * + 1, b = g *, 1 g. Since each word in W is divided into sub-words with a fixed ength, the ength of sub-words in CMP tabe is equa to, and there are * entries in CMP tabe. Note that, G = {X 1 [a..b], X [a..b],, X n [a..b]} denotes a group of sub-words of W, X i W, 1 i n. Next, compute Hamming distance between any two sub-words X i [a..b] and X j [a..b], i j, in the same group by ooing up the CMP tabe, then update the corresponding entry in the INF tabe. Finay, after a comparison processes are finished, chec the vaue of each entry in the INF tabe. If there exists a vaue ower than the parameter of the given constraint, then report that the given set of words does not satisfy the constraint. The detai of the Tabe Looup method for verifying whether a given set of words W satisfies constraint C 1 is described in Agorithm. Note Agorithm can be easiy extended to verify constraint C on any fixed aphabet. * Theorem Agorithm is correct and its time compexity is O( * + n * ) whie W is over binary aphabet.

7 TESTING CODE WORDS WITH CONSTRAINTS 339 Agorithm Tabe Looup Agorithm 1. procedure Tabe Looup (W) {* Output Fai, if the constraint is not satisfied; Success, if the constraint is satisfied *}. Enumerate a possibe sub-words with ength over the binary aphabet to construct the CMP tabe; 3. for g from 1 to do 4. Let X i,g = X i [((g 1) * + 1)..g * ], i = 1,,, n; 5. for i from 1 to n do 6. for j from 1 to n do 7. Looup the vaue of d(x i,g, X j,g ) from the CMP tabe; 8. Let t i,j = t i,j + d(x i,g, X j,g ); 9. end for 10. end for 11. end for 1. for i from 1 to n do 13. for j from 1 to n do 14. if t i,j < 1, where i j then 15. return Fai ; 16. end if 17. end for 18. end for 19. return Success ; 0. end procedure Proof: Correctness: The CMP tabe records Hamming distance between any two possibe sub-words with a fixed ength. Thus we can compare any two sub-words by examining the CMP tabe. Note that, at step 4, each word is divided into sub-words. From steps 3-11, it is obvious to see that the agorithm compares any two sub-words in the same group. For each group, the agorithm adds corresponding vaues to update INF tabe. After comparing a positions of the word, the cumuative vaues in INF tabe correspond to the Hamming distances of any two words. Therefore, Agorithm is correct. * Time compexity: Step costs O( * ) time to construct CMP tabe, since (1) it taes O( ) time to enumerate a possibe sub-words; () to record any two pairs of a possibe sub-words needs * entries; (3) to determine Hamming distance between two sub-words needs O( ) time. In steps 3-11, it is easy to see that it taes On ( * ) time to update INF tabe, and in steps 1-18 it taes O(n ) time to chec whether the vaue of each entry in INF tabe is ess than the parameter of the given constraint. There- * fore, it taes O( * + n * ) time in tota. * Coroary 3 The time compexity of Agorithm is O( α * + n * ), where α is the size of aphabet. Proof: Simiar to the proof of Theorem, the coroary is true. Here, we give an exampe for iustrating the steps in Agorithm. Assume that there nine words X 1,, X 9 in W with = 9. The CMP tabe constructed at step of Agorithm is shown in Fig. 1, and the first iteration of steps 3-11 is depicted in Fig..

8 340 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO Fig. 1. Exampe of CMP tabe. Fig.. Iustration of the steps in Tabe Looup Agorithm and resuts in INF tabe. Next, we appy Tabe Looup method to dea with the verification of constraints C 3, C 4, and C 5. We first consider the cases of verifying constraints C 3 and C 4. Unie the cases of verifying constraints C 1 and C, the agorithm proposed here does not divide a word X p into sub-words, and then to compare them with sub-words of X q. Therefore, INF tabe is not needed here. However, it needs to enumerate a possibe words with ength to construct the CMP tabe. In addition, the agorithm extracts two sub-words X p [1..i] and X q [( i + 1).. ] from X p and X q, respectivey, and concatenates a word Y with the sub-words such that P = X p [1..i] + Y, Q = X q [( i + 1).. ] + Y, and P = Q =. By doing so, the Hamming distance between two extracted sub-words of X p and X q can be obtained from the CMP tabe. Then the agorithm determines whether the constraint is satisfied or not. The detai of the agorithm which is caed Tabe_Shift is shown in Agorithm 3. Agorithm 3 focuses on verifying constraint C 3 and can be easiy extended to verify constraint C 4. Theorem 4 Agorithm 3 is correct and its time compexity is O( * * + n * * ). Proof: Simiar to the proof of Theorem, this agorithm is correct. Time compexity of step in Agorithm 3 is O( * * ) and it taes O(n * * ) in steps Therefore, the time compexity of Agorithm 3 is O( * * + n * * ). For the constraint C 5, the sub-words X a [1..i] and X a [( i + 1).. ] of a word X a for i from to have to be compared with their reverse compement X a [1..i] RC and X a [( i + 1).. ] RC, respectivey, where X a W and a = 1,,, n. As step in Agorithm, it

9 TESTING CODE WORDS WITH CONSTRAINTS 341 Agorithm 3 Tabe_Shift Agorithm 1. procedure Tabe_Shift(W) {* Output Fai, if the constraint is not satisfied; Success, if the constraint is satisfied}. Enumerate a possibe words with ength over the binary aphabet to construct the CMP tabe; 3. for i from + 1 to do 4. Let Y = 11 1, where Y = i; 5. for p from 1 to n do 6. for q from 1 to n do 7. Let P = X p [1..i] + Y and Q = X q [( i + 1).. ] + Y; 8. Looup the vaue of d(p, Q) from the CMP tabe; 9. if d(p, Q) < 3 then 10. return Fai ; 11. end if 1. end for 13. end for 14. end for 15. return Success ; 16. end procedure enumerates a possibe sub-words with ength og over the binary aphabet to construct a og og by CMP tabe. As step 7 in Agorithm 3, it concatenates dummy characters to sub-words X a [1..i], X a [1..i] RC, X a [( i + 1).. ], and X a [( i + 1).. ] RC, so that the ength of the varied sub-words is equa to. Note a sub-word concatenates with dummy characters is caed a varied sub-word. Then it divides each varied sub-word into og sub-words with ength og and oos up CMP tabe to determine whether W satisfies constraint C 5. Simiar to the proof of Theorem and Theorem 4, it taes O( * og ) time to construct CMP tabe and the time compexity of verifying constraint C 5 is O( * og + n* * 5. ENCODING ). og In this section, we utiize the Encoding technique to verify whether a set of words satisfies constraints C 1 C 4. We appy two data structures to speed up the checing. (1) INF tabe (Information tabe): the tabe records the Hamming distance between any two words in the given set of words as described in section 4. () NZO ined ist: the main function of NZO ined-ist is to store the group number of two compared sub-words whie the two sub-words are different. NZO ined ist consists of two inds of nodes: Head nodes and Data nodes. Head node stores the abe (i, j) of any two words X i and X j in data fied and a pointer points to Data nodes, where X i, X j W. Therefore, there are n Head nodes in the ist. Note that, the initia vaue in in the fied of Head node is NULL. Data node stores the group ID, which is defined ater, of two compared subwords if the sub-word of X i and the sub-word of X j are different. The NZO ined ist is iustrated in Fig. 3. In this method, first, each word X i W is divided into * sub-words each with ength. These sub-words are denoted as X i,1, X i,,, X where X = X [a..b], i,g i, i, *

10 34 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO Fig. 3. An iustration of the NZO ined ist. a = (g 1) * + 1, b = g *, 1 g *, and is the parameter of the given con straint. Let G g = {X 1,g, X,g,, X n,g } denote a set of n sub-words, where g is the group ID of G g. Note that, 1 g *. and the tota number of groups is *. Second, using radix sort to sort the sub-words of each group, and then chec if any two sub-words in same group are different. If X i,g and X j,g are different, then update the INF tabe, i.e., t i,j = t i,j + 1, and insert a new Data node with group ID g into the corresponding ocation of the NZO ined ist. Note that the initia vaue of each entry in the INF tabe is 0. Finay, for each entry in the INF tabe, chec if t i,j <. If the vaue t i,j of the entry in the INF tabe is ess than, then scan the corresponding nodes in the NZO ined ist to find the subwords of X i and X j, such that these sub-words are compared again to determine whether d(i, j). To simpify the discussion, et Head(i, j) denote a Head node in which the vaue of data fied is (i, j) and et Data(i, j)[g] denote a Data node, which is ined to Head(i, j) and stores a group ID g in its data fied. The detai of the Encoding method for verifying constraint C 1 is shown in Agorithm 4. The agorithm can easiy be extended to dea with constraint C. We give an exampe to iustrate the steps of Encoding method for verifying constraint C 1. First, we assume that there are eight words, each word has eight bits, and the given parameter =. In steps 3-4, each word is divided into 8* = 4 groups, i.e., G 1,, G 4, as shown in Fig. 4 (a). And Fig. 4 (b) shows the steps 5-13 in the Encoding method. If the vaue of t i, j in INF tabe is ess than, then according to NZO ined ist, we now which sub-words need to be compared again to determine whether the d(i, j). From the INF tabe and NZO ined ist shown in Fig. 5, it is nown that two pairs of sub-words X 1 [1..] and X 3 [1..], X 7 [5..6] and X 3 [5..6] need to be compared again in this exampe. Theorem 5 Agorithm 4 is correct and its time compexity is On ( * * ). Proof: First, we prove the correctness of the agorithm. The vaue in the INF tabe corresponding to two words X i and X j ony be added when the sub-words of X i and X j are different. Therefore, if this vaue is greater or equa to, it means d(x i, X j ) must be greater than or equa to, where is the parameter of the given constraint. Our agorithm checs each entry of the INF tabe to mae sure that any two words satisfy the constraint. However, there may be some entries with vaue ess than, and the sub-words of these entries need to be compared again. To compare those sub-words, the agorithm utiizes the NZO ined ist to eep the information of those sub-words as step 10, such that the sub-words can be easiy extracted from the words. Therefore, our agorithm is correct. Second, we

11 TESTING CODE WORDS WITH CONSTRAINTS 343 Agorithm 4 Encoding Agorithm 1. procedure Encoding (W) {* Output Fai, if the constraint is not satisfied; Success, if the constraint is satisfied *}. for g from 1 to * 1 do 3. Let X i,g = X i [((g 1) * 1).. * + g ], i= 1,, K, n ; Let G g = {X 1,g, X,g,, X n,g }; 5. Perform radix sort on G g ; 6. for i from 1 to n do 7. for j from 1 to n do 8. if X i,g X j,g, where i j then 9. t i,j = t i,j + 1; {* Update INF tabe *} 10. Create a new Data node: Data(i, j)[g] and insert the node into the ocation after Head(i, j) in the NZO ined ist 11. end if 1. end for 13. end for 14. end for 15. for i from 1 to n do 16. for j from 1 to n do 17. if t i,j < 1 then 18. Find the Head node Head(i, j) and et d = 0; 19. whie The in fied of Head(i, j) NULL do 0. Let g be the vaue that stored in the data fied of a Data node, which is ined after Head(i, j), i.e., Data(i, j)[g ], then remove this Data node; 1. Compute Hamming distance between X i,g and X j,g ;. Let d = d + d(x i,g, X j,g ); 3. end whie 4. if d < 1 then 5. return Fai 6. end if 7. end if 8. end for 9. end for 30. return Success 31. end procedure anayze the time compexity of the agorithm. Each word in the set is partitioned into * sub-words each with ength. After radix sort, it compares any two sub-words, and then updates the INF tabe and adds new node into the NZO ined ist. Thus it taes On ( * * ) time in steps -14. If there is any entry in the INF tabe with vaue ess than, then the number of groups which is stored in corresponding ocation of the NZO ined ist must be ess than. Therefore, using information stored in the entries of the NZO ined ist, the agorithm in steps 15-9 compares the sub-words again to determine whether the d(i, j) in On ( * * ) time. Hence, it taes On ( * * ) time in tota.

12 344 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO (a) (b) Fig. 4. (a) Each word is divided into * sub-words each with ength ; (b) An iustration of steps for updating the INF tabe. ING Tabe NZO ined ist Fig. 5. Reationship between INF tabe and NZO ined ist. Note that, the Encoding technique can aso be used to verify constraints C 3 and C 4. It first concatenates i dummy characters to sub-words X a [1..i] and X a [( i + 1).. ], for i from + 1 to, where X a W, a = 1,, n. Then, simiar to Agorithm 4, it divides the varied sub-words into sub-words each with ength * and uses INF tabe and NZO ined ist to perform the testing. It is easy to see that the time compexity of verifying constraints C 3 and C 4 is On ( * * * ), where is the given parameter of the corresponding constraint. 6. CONCLUDING REMARKS We have proposed three techniques for soving the verification of the given set of words. The first is Enumeration method which can be appied to constraints C 1 C 4 whie og the parameters of the given constraints in beow ( n O og ogn ). The second is Tabe Looup method which can be appied to constraints C 1 C 5. The Tabe Looup method is an appropriate approach whie the ength of a word and the given parameter of a constraint are in the same order O(og n). Finay, the Encoding method can aso be appied to C 1 C 4 whie the ength of a word is greater than the order O(og n). The proposed approaches determine whether a set of words satisfy a set of combinatoria constraints faster than the trivia soution.

13 TESTING CODE WORDS WITH CONSTRAINTS 345 REFERENCES 1. M. Arita, A. Nishiawa, M. Hagiya, K. Komiya, H. Gouzu, and K. Saamoto, Improving sequence design for DNA computing, in Proceedings of the Genetic and Evoutionary Computation Conference, 000, pp A. Brenneman and A. E. Condon, Strand design for bio-moecuar computation, Theoretica Computer Science, Vo. 87, 001, pp R. Deaton, R. C. Murphy, M. Garzon, D. R. Franceschetti, and S. E. Stevens Jr., Reiabiity and efficiency of a DNA-based computation, Physica Review Letters, Vo. 8, 1998, pp J. K. Gaant, String compression agorithms, Ph.D. dissertation, Department of Eectrica Engineering and Computer Science, Princeton University, Princeton, NJ, M. Garzon, R. Deaton, P. Neathery, R. C. Murphy, S. E. Stevens Jr., and D. R. Franceschetti, A new metric for DNA computing, in Proceedings of Genetic Programming, 1997, pp D. A. Huffman, A method for the construction of minimum-redundancy codes, in Proceedings of the Institute of Radio Engineers, Vo. 40, 1951, pp M. Y. Kao, M. Sanghi, and R. Schweer, Randomized fast design of short DNA words, Lecture Notes in Computer Science, Vo. 3580, 005, pp A. Marathe, A. Condon, and R. M. Corn, On combinatoria DNA word design, Journa of Computationa Bioogy, Vo. 8, 001, pp T. Martinetz, J. E. Gewehr, and J. T. Kim, Statistica earning for detecting protein-dna-binding sites, in Proceedings of the Internationa Joint Conference on Neura Networs, Vo. 4, 003, pp W. Shi and W. Zhou, Identifying transcription factor binding sites in promoters of human genes, in Proceedings of Internationa Muti-Conference on Computing in the Goba Information Technoogy, 006, pp S. Y. Shin, D. Kim, I. H. Lee, and B. T. Zhang, Evoutionary sequence generation for reiabe DNA computing, in Proceedings of Congress on Evoutionary Computation, 00, pp F. Tanaa, M. Naatsugawa, M. Yamamoto, T. Shiba, and A. Ohuchi, Deveoping support system for sequence design in DNA computing, in Proceedings of the 7th Internationa Worshop on DNA Based Computers, 00, pp J. Vitter, Design and anaysis of dynamic Huffman codes, Journa of the Association for Computing Machinery, Vo. 34, 1987, pp E. H. Yang and Z. Zhang, The shortest common superstring probem: Average case anaysis for both exact and approximate matching, IEEE Transactions on Information Theory, Vo. 45, 1999, pp Hsin-Wen Wei ( ) received the B.S. degree in Information and Computer Engineering from Chung Yuan Christian University in 000. She is currenty a Ph.D. candidate in Computer Science at Nationa Tsing Hua University in Taiwan. Her research interests are in scheduing theory, rea-time and embedded computer systems, and graph theory.

14 346 HSIN-WEN WEI, WAN-CHEN LU, PEI-CHI HUANG, WEI-KUAN SHIH AND MING-YANG KAO Wan-Chen Lu ( ) received the B.S. degree in Information Management from Chang Gang University in 00, and the Ph.D. degree in Computer Science from Nationa Tsing Hua University in 007. Her research interests are scheduing theory, rea-time systems, mutimedia systems, and graph theory. Pei-Chi Huang ( ) received the B.S. degree in Industria Technoogy Education from Nationa Taiwan Norma University in 004, and M.S. degree in Computer Science from Nationa Tsing Hua University in 005. She is currenty a Ph.D. candidate in Computer Science at Nationa Tsing Hua University in Taiwan. Her research interests are scheduing theory, rea-time systems, and graph theory. Wei-Kuan Shih ( ) received the B.S. and M.S. degrees in Computer Science from the Nationa Taiwan University, and the Ph.D. degree in Computer Science from the University of Iinois, Urbana-Champaign. From 1986 to 1988, he was with the Institute of Information Science, Academia Sinica, Taiwan. He is a Professor in the Department of Computer Science at the Nationa Tsing Hua University, Taiwan. His research interests are rea-time systems, VLSI design automation, and wireess communication. Ming-Yang Kao ( ) received the Ph.D. degree in Computer Science from Yae University in He is a Professor in the Department of Eectrica Engineering and Computer Science at Northwestern University, Evanston, IL. His research interests incude appications, design, anaysis, and impementation of agorithms.