Counting Parameterized Border Arrays for a Binary Alphabet

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1 Counting Parameterized Border Array for a Binary Alphabet Tomohiro I 1, Shunuke Inenaga 2, Hideo Bannai 1, and Maayuki Takeda 1 1 Department of Informatic, Kyuhu Univerity 2 Graduate School of Information Science and Electrical Engineering, Kyuhu Univerity 744 Motooka, Nihiku, Fukuoka, Japan. {tomohiro.i,bannai,takeda}@i.kyuhu-u.ac.jp inenaga@c.cce.kyuhu-u.ac.jp Abtract. The parameterized pattern matching problem i a kind of pattern matching problem, where a pattern i conidered to occur in a text when there exit a renaming bijection on the alphabet with which the pattern can be tranformed into a ubtring of the text. A parameterized border array (p-border array) i an analogue of a border array of a tandard tring, which i alo known a the failure function of the Morri-Pratt pattern matching algorithm. In thi paper we preent a linear time algorithm to verify if a given integer array i a valid p-border array for a binary alphabet. We alo how a linear time algorithm to compute all binary parameterized tring haring a given p-border array. In addition, we give an algorithm which compute all p-border array of length at mot n, where n i a a given threhold. Thi algorithm run in time linear in the number of output p-border array. 1 Introduction 1.1 Parameterized Matching and Parameterized Border Array The parameterized matching (p-matching) problem [1] i a kind of tring matching problem, where a pattern i conidered to occur in a text when there exit a renaming bijection on the alphabet with which the pattern can be tranformed into a ubtring of the text. A parameterized tring (p-tring) i formally an element of (Π Σ), where Π i the et of parameter ymbol and Σ the et of contant ymbol. The renaming bijection ued in p-matching are the identity on Σ, that i, every contant ymbol X Σ i mapped to X, while ymbol in Π can be interchanged. Parameterized matching ha application in oftware maintenance [2, 1], plagiarim detection [3], and RNA tructural matching [4], thu it ha been extenively tudied in the lat decade [5 12]. Of variou efficient method olving the p-matching problem, thi paper focue on the algorithm of Idury and Schäffer [13] that olve the p-matching problem for multiple pattern. Their algorithm modifie the Aho-Coraick automata [14], replacing the goto and fail function with the pgoto and pfail function, repectively. When the input i a ingle pattern p-tring of length m, the

2 2 T. I, S. Inenaga, H. Bannai, and M. Takeda pfail function can be implemented by an array of length m, and we call the array the parameterized border array (p-border array) of the pattern p-tring, which i the parameterized verion of the border array [15]. The p-border array of a given pattern can be computed in linear time [13]. 1.2 Revere and Enumerating Problem on String The revere problem for tandard border array [15] wa firt introduced by Franěk et al. [16]. They propoed a linear time algorithm to verify if a given integer array i the border array of ome tring. Their algorithm work for both bounded and unbounded alphabet. Duval et al. [17] propoed a impler algorithm to olve the ame problem in linear time for bounded alphabet. Moore et al. [18] preented an algorithm to enumerate all border array of length at mot n, where n i a given poitive integer. They propoed a notion of b-equivalence of tring uch that two tring are b-equivalent if they have the ame border array. The lexicographically mallet one of each b-equivalence cla i called b-canonical tring of the cla. Their algorithm i alo able to output all b-canonical tring of length up to a given integer n. Franěk et al. [16] pointed out that the time complexity analyi of [18] i incorrect, and howed a new algorithm which olve the ame problem in O(b n ) time uing O(b n ) pace, where b n denote the number of border array of length at mot n. The revere problem for ome other tring data tructure, uch a uffix array [19], directed acyclic word graph [20], directed acyclic ubequence graph [21] have been olved in linear time [22, 23]. The problem of enumerating all uffix array wa conidered in [24]. An algorithm to enumerate all p-ditinct tring wa propoed in [18], where two tring are aid to be p-ditinct if they do not parameterized-match. 1.3 Our Contribution Thi paper conider the reveral of the problem of computing the p-border array of a given pattern p-tring. That i, given an integer array α, determine if there exit a p-tring whoe p-border array i α. In thi paper, we preent a linear time algorithm which olve the above problem for a binary parameter alphabet ( Π = 2). We then conider a more challenging problem: given a poitive integer n, enumerate all p-border array of length at mot n. We propoe an algorithm that olve the enumerating problem in O(B n ) time for a binary parameter alphabet, where B n i the number of all p-border array of length n for a binary parameter alphabet. We alo give a imple algorithm to output all tring which hare the ame p-border array. A p-border i a dual concept of a parameterized period of a p-tring. Apotolico and Giancarlo [11] howed that a complete analogy to the weak periodicity lemma [25] tand for p-tring over a binary alphabet. Our reult reveal yet another imilarity of p-tring over a binary alphabet and tandard tring in term of periodicity.

3 2 Preliminarie Counting Parameterized Border Array for a Binary Alphabet Parameterized String Matching Let Σ and Π be two dijoint finite et of contant ymbol and parameter ymbol, repectively. An element of (Σ Π) i called a p-tring. The length of any p-tring i the total number of contant and parameter ymbol in and i denoted by. For any p-tring of length n, the i-th ymbol i denoted by [i] for each 1 i n, and the ubtring tarting at poition i and ending at poition j i denoted by [i : j] for 1 i j n. In particular, [1 : j] and [i : n] denote the prefix of length j and the uffix of length n i + 1 of, repectively. Any two p-tring and t of the ame length m are aid to parameterized match if can be tranformed into t by applying a renaming function f from the ymbol of to the ymbol of t, uch that f i the identity on the contant alphabet. For example, let Π = {a, b, c}, Σ = {X, Y}, = abcxaby and t = bcaxbcy. We then have t with the renaming function f uch that f(a) = b, f(b) = c, f(c) = a, f(x) = X, and f(y) = Y. We write t when and t p-match. Amir et al. [5] howed that we have only to conider p-tring over Π when conidering p-matching. Lemma 1 ([5]). The p-matching problem on alphabet Σ Π i reducible in linear time to the p-matching problem on alphabet Π. 2.2 Parameterized Border Array A in the cae of tandard tring matching, we can define the parameterized border (p-border) and the parameterized border array (p-border array). Definition 1. A parameterized border ( p-border) of a p-tring of length n i any integer j uch that 0 j < n and [1 : j] [n j + 1 : n]. For example, the et of p-border of p-tring aabbaa i {4, 2, 1, 0}, ince aabb bbaa, aa aa, a a, and ε ε. Definition 2. The parameterized border array ( p-border array) β of any p- tring of length n i an array of length n uch that β [i] = j, where j i the longet p-border of [1 : i]. For example, the p-border array of p-tring aabbaa i [0, 1, 1, 2, 3, 4]. When it i clear from the context, we abbreviate β a β. The p-border array β of p-tring wa firt explicitly introduced by Idury and Schäffer [13] a the pfail function, where the pfail function i ued in their Aho-Coraick [14] type algorithm that olve the p-matching problem for multiple pattern. Given a pattern p-tring p of length m, the p-border array β p can be computed in O(m log Π ) time, and the p-matching problem can be olved in O(n log Π ) time for any text p-tring of length n.

4 4 T. I, S. Inenaga, H. Bannai, and M. Takeda 2.3 Problem Thi paper deal with the following problem. Problem 1 (Verifying valid p-border array). Given an integer array α of length n, determine if there exit a p-tring uch that β = α. Problem 2 (Computing all p-tring haring the ame p-border array). Given an integer array α which i a valid p-border array, compute every p-tring uch that β = α. Problem 3 (Computing all p-border array). Given a poitive integer n, compute all p-border array of length at mot n. In the following ection, we give efficient olution to the above problem for a binary alphabet, that i, Π = 2. 3 Algorithm Thi ection preent our algorithm which olve Problem 1, Problem 2 and Problem 3 for the cae Π = 2. We begin with the baic propoition on p-border array. Propoition 1. For any p-border array β[1..i] of length i 2, β[1..i 1] i a p-border array of length i 1. Proof. Let be any p-tring uch that β = β. It i clear from Definition 2 that β [1..i 1] i the p-border array of the p-tring [1 : i 1]. Due to the above propoition, given an integer array α[1..n], we can check if it i a p-border array of ome tring of length n by teting each element of α in increaing order (from 1 to n). If we find any 1 i n uch that α[1..i] i not a p-border array of length i, then α[1..n] can never be a p-border of length n. In what follow, we how how to check each element of a given integer array in increaing order. For any p-border array β of length n and any integer 1 i n, let { β k β[i] if k = 1, [i] = β[β k 1 [i]] if k > 1 and β k 1 [i] 1. It follow from Definition 2 that the equence i, β[i], β 2 [i],... i monotone decreaing to zero, hence finite. Lemma 2. For any p-tring of length i, {β 1 [i], β 2 [i],..., 0} i the et of the p-border of.

5 Counting Parameterized Border Array for a Binary Alphabet 5 Proof. Firt we how by induction that for every k, 1 k k, β k [i] i a p- border of, where k i the integer uch that β k [i] = 0. By Definition 2, β[i] 1 i the longet p-border of. Suppoe that for ome k, 1 k < k, β k [i] i a p-border of. Here β k+1 [i] i the longet p-border of β k [i]. Let f and g be the bijection uch that Since f([1])f([2]) f([β k [i]]) = [i β k [i] + 1 : i], g([1])g([2]) g([β k+1 [i]]) = [β k [i] β k+1 [i] + 1 : β k [i]]. f(g([1]))f(g([2])) f(g([β k+1 [i]])) = f([β k [i] β k+1 [i] + 1])f([β k [i] β k+1 [i] + 2]) f([β k [i]]) = [i β k+1 [i] + 1 : i], we obtain [1 : β k+1 [i]] [i β k+1 [i] + 1 : i]. Hence β k+1 [i] i a p-border of. We now how any other j i not a p-border of. Aume for contrary that j, β k+1 Since [i] < j < β k [i], i a p-border of. Let q be the bijection uch that q([i j + 1])q([i j + 2]) q([i]) = [1 : j]. q(f([β k [i] j + 1]))q(f([β k [i] j + 2])) q(f([β k [i]])) = q([i j + 1])q([i j + 2]) q([i]) = [1 : j], we obtain [1 : j] [β k [i] j + 1 : β k [i]]. Hence j i a p-border of [1 : β k [i]]. However thi contradict with the definition that β k+1 [i] i the longet p-border of [1 : β k [i]]. Lemma 3. For any p-tring of length i 1 and a Π, every p-border of a i an element of the et {β 1 [i] + 1, β 2 [i] + 1,..., 1}. Proof. Aume for contrary that a ha a p-border j + 1 / {β[i] 1 + 1, β[i] 2 + 1,..., 1}. Since [1 : j + 1] [i j + 1 : i]a, we have [1 : j] [i j + 1 : i] and j i a p-border of. It follow from Lemma 2 that j {β[i], 1 β[i], 2..., 0}. Thi contradict with the aumption. Baed on Lemma 2 and Lemma 3, we can efficiently compute the p-border array β of a given p-tring. Alo, our algorithm to olve Problem 1 i baed on thee lemma. Note that Propoition 1, Lemma 2 and Lemma 3 hold for p-tring over Π of arbitrary ize. In the equel we how how to elect m {β 1 [i] + 1, β 2 [i] + 1,..., 1} uch that β [1..i]m i a valid p-border array of length i + 1. The following propoition, lemma and theorem hold for a binary parameter alphabet, Π = 2. For p-border array of length at mot 2, we have the next propoition.

6 6 T. I, S. Inenaga, H. Bannai, and M. Takeda i a f β h-1[i]+1 g f β h[i]+1 Fig. 1. Illutration for Lemma 5. Propoition 2. For any p-tring of length 1, β [1] = 0. For any p-tring of length 2, β [2] = 1. Proof. Let Π = {a, b}. It i clear that the longet p-border of a and b i 0. The p-tring of length 2 over Π are aa, ab, ba, and bb. Obviouly the longet p-border of each of them i 1. For p-border array of length more than 2, we have the following lemma. Lemma 4. For any p-tring Π, if j 2 i a p-border of a with a Π, then j i not a p-border of b, where b Π {a}. Proof. Aume for contrary that j i a p-border of b. Then, let f and g be the bijection on Π uch that f([1])f([2]) f([j]) = [i j + 2 : i]a, g([1])g([2]) g([j]) = [i j + 2 : i]b. We get from f([1])f([2]) f([j 1]) = [i j+2 : i] = g([1])f([2]) g([j 1]) that f and g are the ame bijection. However, f([j]) = a b = g([j]) implie that f and g are different bijection, a contradiction. Hence j i not a p-border of b. Lemma 5. For any p-tring of length i, if β [β h 1 [i] + 1] = β h [i] + 1 and [i] + 1 i a p-border of a with a Π, then β h [i] + 1 i a p-border of a. β h 1 (See alo Fig. 1.) Proof. Let f and g be the bijection on Π uch that Since f([1])f([2]) f([β h 1 [i] + 1]) = [i β h 1 [i] + 1 : i]a, g([1])g([2]) g([β h [i] + 1]) = [β h 1 [i] β h [i] + 1 : β h 1 [i] + 1]. f(g([1]))f(g([2])) f(g([β h [i] + 1])) = f([β h 1 [i] β h [i] + 1])f([β h 1 [i] β h [i] + 2]) f([β h 1 [i] + 1]) = [i β h [i] + 1 : i]a,

7 Counting Parameterized Border Array for a Binary Alphabet 7 i a f β h-1[i]+1 q f β h[i] q f a q f b Fig. 2. Illutration for Lemma 6. we obtain [1 : β h [i] + 1] [i β h [i] + 1 : i]a. Hence β h [i] + 1 i a p-border of a. Lemma 6. For any p-tring of length i, if β [β h 1 [i] + 1] β h [i] + 1 and [i] + 1 i a p-border of a with a Π, then β h [i] + 1 i a p-border of b uch β h 1 that b Π {a}. (See alo Fig. 2.) Proof. Let f and g be the bijection on Π uch that f([1])f([2]) f([β h 1 [i] + 1]) = [i β h 1 [i] + 1 : i]a, q([1])q([2]) q([β h [i]]) = [β h 1 [i] β h [i] + 1 : β h 1 Becaue β [β h 1 [i] + 1] β h [i] + 1, we know that q([β h [i] + 1]) [β h 1 [i] + 1]. Since f([β h 1 [i]+1]) = a and Π = {a, b}, f(q([β h [i]+1])) = b. Hence β h [i]+1 i a p-border of b. The following i a key lemma to olving our problem. Lemma 7. For any p-border array β of length i 2, β[1..i]m 1 and β[1..i]m 2 are the p-border array of length i + 1, where m 1 = β[i] + 1 and β l if β[β l 1 [i] + 1] β l [i] + 1 for ome 1 < l < k and [i] + 1 m 2 = β[β h 1 [i] + 1] = β h [i] + 1 for any 1 < h < l, 1 otherwie, where k i the integer uch that β k [i] = 0. Proof. Conider any p-tring of length i uch that β = β. By definition, there exit a bijection f on Π uch that f([1])f([2]) f([β[i]]) = [i β[i]+1 : i]. Let a = f([β[i]+1]). Then f([1])f([2]) f([β[i]])f([β[i]+1]) = [i β[i]+1 : [i]].

8 8 T. I, S. Inenaga, H. Bannai, and M. Takeda Algorithm 1: Algorithm to olve Problem Input: α[1..n] : a given integer array Output: return whether α i a valid p-border array or not if α[1..2] [0, 1] then return invalid; for i = 3 to n do if α[i] = α[i 1] + 1 then continue; d α[i 1]; d α[d ]; while d > 0 & d + 1 = α[d + 1] do d d; d α[d ]; if α[i] = d + 1 then continue; return invalid; return valid; i]a. Note that β[1..i](β[i] + 1) i the p-border array of a becaue a can have no p-border longer than β[i] + 1. It follow from Lemma 5 that β h [i]+1 i a p-border of a. Then, by Lemma 6, β l [i] + 1 i a p-border of b. Since β h [i] 1, by Lemma 4, β h [i] + 1 i not a p- border of b. Hence β l [i] + 1 i the longet p-border of b. We are ready to tate the following theorem. Theorem 1. Problem 1 can be olved in linear time for a binary parameter alphabet. Proof. Algorithm 1 decribe the operation to olve Problem 1. Given an integer array of length n, the algorithm firt check if α[1..2] = [0, 1] due to Propoition 2. If α[1..2] = [0, 1], then for each i = 3,..., n (in increaing order) the algorithm check whether α[i] atifie one of the condition of Lemma 7. The time analyi i imilar to that of Theorem 2.3 of [16]. In each iteration of the for loop, the value of d increae by at mot 1. However, each execution of the while loop decreae the value of d by at leat 1. Hence the total time cot of the for loop i O(n). Theorem 2. Problem 2 can be olved in linear time for a binary parameter alphabet. Proof. It follow from Propoition 2 that the p-border array of all p-tring of length 2 (aa, ab, ba, and bb) i [0, 1]. By Propoition 1, for any p-border array β[1..n] with n 2, we have β[1..2] = [0, 1]. Hence each p-border array β[1..n] with n 2 correpond to exactly four p-tring each of which begin with aa, ab, ba, and bb, repectively. Algorithm 2 i an algorithm to olve Problem 2. Technically x aa can be computed by aa [β[i]] xor aa [β[i]+1] xor aa [i] on binary alphabet Π = {0, 1}. Hence thi counting algorithm work in linear time.

9 Counting Parameterized Border Array for a Binary Alphabet 9 Algorithm 2: Algorithm to compute all p-tring haring the ame p- border array Input: β[1..n] : a p-border array Output: all p-tring haring the ame p-border array β[1..n] aa aa; ab ab; bb bb; ba ba; for i = 3 to n do Let f be the bijection on Π.t. f( aa [β[i]]) = aa [i]; Let g be the bijection on Π.t. g( ab [β[i]]) = ab [i]; x aa f( aa [β[i] + 1]); x ab g( ab [β[i] + 1]); x aa y Π {x aa }; x ab z Π {x ab }; if β[i] = β[i 1] + 1 then aa [i] x aa ; ab [i] x ab ; bb [i] x aa; ba [i] x ab ; ele aa[i] x aa; ab [i] x ab ; bb [i] x aa ; ba [i] x ab ; end Output: aa [1 : n], ab [1 : n], bb [1 : n], ba [1 : n] ( 0 ) ( 1 ) ( 1 ) ( 2 ) ( 1 ) ( 2 ) ( 1 ) ( 3 ) Fig. 3. The tree T 4 which repreent all p-border array of length at mot 4 for a binary alphabet. We now conider Problem 3. By Propoition 1 and Lemma 7, computing all p-border array of length at mot n can be accomplihed uing a rooted tree tructure T n of height n 1. Each node of T n of height i 1 correpond to an integer j uch that j i the longet p-border of ome p-tring of length i over a binary alphabet, hence the path from the root to that node repreent the p-border array of the p-tring. Fig. 3 repreent T 4. Theorem 3. Problem 3 can be olved in O(B n ) time for a binary parameter alphabet, where B n denote the number of p-border array of length n. Proof. Propoition 2 and Lemma 7 imply that every internal node of T n of height at leat 1 ha exactly two children. Hence the total number of node of T n i O(B n ). We compute T n in a depth-firt manner. Algorithm 3 how a function that compute the children of a given node of T n. It i not difficult to ee that

10 10 T. I, S. Inenaga, H. Bannai, and M. Takeda Algorithm 3: Function to compute the children of a node of T n Input: i : length of the current p-border array, 2 i n Reult: compute the children of the current node // β[1..n] i allocated globally and β[1..i] repreent the current p-border array. function getchildren(i) if i = n then return ; β[i + 1] β[i] + 1; report β[i + 1]; getchildren(i + 1); d β[i]; d β[d ]; while d > 0 & d + 1 = β[d + 1] do d d; d β[d ]; β[i + 1] d + 1; report β[i + 1]; getchildren(i + 1); return ; each child of a given node can be computed in amortized contant time. Hence Problem 3 can be olved in O(B n ) time for a binary parameter alphabet. We remark that if each p-border array in T n can be dicarded after it i generated, then we can compute all p-border array of length at mot n uing O(n) pace. Since every internal node of T n of height at leat 1 ha exactly two children and the root ha one child, B n = 2 n 2 for n 2. Thu the pace requirement can be reduced to O(log B n ). 4 Concluion and Open Problem A parameterized border array (p-border array) i a ueful data tructure for parameterized pattern matching. In thi paper, we preented a linear time algorithm which tet if a given integer array i a valid p-border array for a binary alphabet. We alo gave a linear time algorithm to compute all binary p-tring that hare a given p-border array. Finally, we propoed an algorithm which compute all p-border array of length at mot n, where n i a given threhold. Thi algorithm work in O(B n ) time, where B n denote the number of p-border array of length n for a binary alphabet. Problem 1,2, and 3 are open for a larger alphabet. To ee one of the reaon of why, we how that Lemma 4 doe not hold for a larger alphabet. Conider a p-tring = abac over Π = {a, b, c}. Oberve that β = [0, 1, 2, 2]. Although β [4] = 2 i a p-border of abac, it i alo a p-border of another p-tring abab ince ab ab. Hence Lemma 4 doe not hold if Π 3. Our future work alo include the following:

11 Counting Parameterized Border Array for a Binary Alphabet 11 Verify if a given integer array i a parameterized uffix array [12]. Compute all parameterized uffix array of length at mot n. In [12], a linear time algorithm which directly contruct the parameterized uffix array for a given binary tring wa propoed. Thi algorithm might be ued a a bai for olving the above problem regarding parameterized uffix array. Reference 1. Baker, B.S.: Parameterized pattern matching: Algorithm and application. Journal of Computer and Sytem Science 52(1) (1996) Baker, B.S.: A program for identifying duplicated code. Computing Science and Statitic 24 (1992) Fredrikon, K., Mozgovoy, M.: Efficient parameterized tring matching. Information Proceing Letter 100(3) (2006) Shibuya, T.: Generalization of a uffix tree for RNA tructural pattern matching. Algorithmica 39(1) (2004) Amir, A., Farach, M., Muthukrihnan, S.: Alphabet dependence in parameterized matching. Information Proceing Letter 49(3) (1994) Baker, B.S.: Parameterized pattern matching by Boyer-Moore-type algorithm. In: Proc. 6th annual ACM-SIAM Sympoium on Dicrete Algorithm (SODA 95). (1995) Koaraju, S.: Fater algorithm for the contruction of parameterized uffix tree. In: Proc. 36th Annual Sympoium on Foundation of Computer Science (FOCS 95). (1995) Hazay, C., Lewentein, M., Tur, D.: Two dimenional parameterized matching. In: Proc. 16th Annual Sympoium on Combinatorial Pattern Matching (CPM 05). Volume 3537 of Lecture Note in Computer Science. (2005) Hazay, C., Lewentein, M., Sokol, D.: Approximate parameterized matching. ACM Tranaction on Algorithm 3(3) (2007) Article No Apotolico, A., Erdö, P.L., Lewentein, M.: Parameterized matching with mimatche. Journal of Dicrete Algorithm 5(1) (2007) Apotolico, A., Giancarlo, R.: Periodicity and repetition in parameterized tring. Dicrete Applied Mathematic 156(9) (2008) Deguchi, S., Higahijima, F., Bannai, H., Inenaga, S.,, Takeda, M.: Parameterized uffix array for binary tring. In: Proc. The Prague Stringology Conference 08 (PSC 08). (2008) Idury, R.M., Schäffer, A.A.: Multiple matching of parameterized pattern. Theoretical Computer Science 154(2) (1996) Aho, A.V., Coraick, M.J.: Efficient tring matching: An aid to bibliographic earch. Communication of the ACM 18(6) (1975) Morri, J.H., Pratt, V.R.: A linear pattern-matching algorithm. Technical Report Report 40, Univerity of California, Berkeley (1970) 16. Franek, F., Gao, S., Lu, W., Ryan, P.J., Smyth, W.F., Sun, Y., Yang, L.: Verifying a border array in linear time. J. Combinatorial Math. and Combinatorial Computing 42 (2002) Duval, J.P., Lecroq, T., Lefevre, A.: Border array on bounded alphabet. Journal of Automata, Language and Combinatoric 10(1) (2005) 51 60

12 12 T. I, S. Inenaga, H. Bannai, and M. Takeda 18. Moore, D., Smyth, W., Miller, D.: Counting ditinct tring. Algorithmica 23(1) (1999) Manber, U., Myer, G.: Suffix array: a new method for on-line tring earche. SIAM J. Computing 22(5) (1993) Blumer, A., Blumer, J., Hauler, D., Ehrenfeucht, A., Chen, M.T., Seifera, J.: The mallet automaton recognizing the ubword of a text. Theoretical Computer Science 40 (1985) Baeza-Yate, R.A.: Searching ubequence (note). Theoretical Computer Science 78(2) (1991) Duval, J.P., Lefebvre, A.: Word over an ordered alphabet and uffix permutation. Theoretical Informatic and Application 36 (2002) Bannai, H., Inenaga, S., Shinohara, A., Takeda, M.: Inferring tring from graph and array. In: Proc. 28th International Sympoium on Mathematical Foundation of Computer Science (MFCS 2003). Volume 2747 of Lecture Note in Computer Science. (2003) Schürmann, K.B., Stoye, J.: Counting uffix array and tring. Theoretical Computer Science 395(2-1) (2008) Lyndon, R.C., Schützenberger, M.P.: The equation a M = b N c P in a free group. Michigan Math. J. 9(4) (1962)

Codes Correcting Two Deletions

Codes Correcting Two Deletions 1 Code Correcting Two Deletion Ryan Gabry and Frederic Sala Spawar Sytem Center Univerity of California, Lo Angele ryan.gabry@navy.mil fredala@ucla.edu Abtract In thi work, we invetigate the problem of