PRIMITIVITY ON PARTIAL ARRAYS

Transcription

1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu PRIMITIVITY ON PARTIAL ARRAYS S. Vijayachitra 1, K. Sasikala 2, T. Kalyani 3, D.G. Thomas 4 1 Sathyabama University, Chennai 2 Department of Mathematics St Joseph s College of Engineering, Chennai 3 Department of Mathematics St Joseph s Institute of Technology, Chennai 4 Department of Mathematics Madras Christian College, Tambaram Abstract A partial word is a word that contains some holes known as do not symbols and such places can be replaced by any letter from the underlying alphabet. Research in combinatorics on partial words is underway [2, 3, 4, 5] and has impact in the areas like molecular biology, nano technology and in particular for finding good encodings in DNA computations. In this paper we extend the fundamental property called primitivity on partial words to partial arrays. AMS Subject Classification: 68R15. Key Words and Phrases: Partial words, partial arrays, periods of partial arrays and primitive partial words. 1 Introduction In 1999 Partial words were introduced by J. Berstel and L. Boasson in the context of gene comparison. Alignment of two genes can be ijpam.eu

2 viewed as a construction of two partial words that are said to be compatible.while a word can be described by a total function, a partial word can be described by a partial function. A two dimensional string is called a picture or an array and is defined as a rectangular array of symbols taken from a finite alphabet. The study of repetitions of word or partial word is very important both from the theoretical and the application point of view. A partial array A of size (m, n) over Σ, a non empty set or an alphabet is a partial function Z+ 2 Σ from the set of all positive integers to Σ. This paper introduces the primitivity on partial arrays and discuss about the fundamental properties of primitivity on partial arrays. 2 Preliminaries This section is devoted to review the basic concepts of partial words, partial arrays and periodicity on partial arrays. Partial Words Definition 1. A partial word u of length n over A, a nonempty finite alphabet, is a partial map u : {1, 2,..., n} A. If 1 i n then i belongs to the domain of u (denoted by Domain(u)) in the case where u(i) is defined and i belongs to the set of holes of u (denoted by Hole(u)) otherwise. A word [1, 3] is a partial word over A with an empty set of holes. Definition 2. Let u be a partial word of length n over A. The companion of u (denoted by u ) is the map u : {1, 2,..., n} A { } defined by { u(i) if i Domain(u) u (i) = otherwise The symbol of is viewed as a do not know symbol. The word u = ba ab is the companion of the partial word. The length of the partial word is 6. D(u) = {1, 2, 4, 5}. H(u) = {3, 6}. Let u and v be wo partial words of length n. The partial word u is said to be contained in the partial word v (denote by u v) if ijpam.eu

3 Domain(u) Domain(v) and u(i) = v(i) for all i Domain(u). The partial words u and v are called compatible (denoted by u v) if there exists a partial word w such that u w and v w (in which case we define u v by u u v and v u v and Domain(u v) = Domain(u) Domain(v)). As an example, u = aba a and v = abab a are the companians of two partial words u and v that are compatible and (u v) = abab a. A partial word u is primitive if there exists no word v such that u v n with n 2. Partial Arrays Definition 3. A partial array A of size (m, n) over Σ, a nonempty set or an alphabet, is a partial function Z+ 2 Σ where Z + is the set of all positive integers. For 1 i m, 1 j n if A(i, j) is defined then we say that (i, j) belongs to the domain of A (denoted by (i, j) D(A)). Otherwise we say that (i, j) belongs to the set of holes of A (denoted by (i, j) H(A)). An array [6] over Σ is a partial array over Σ with an empty set of holes. Definition 4. If A is a partial array of size (m, n) over Σ, then the companion of A (denoted by A ) is the total function A : Z+ 2 Σ { } defined by { A(i, j) if (i, j) D(A) A (i, j) = otherwise where Σ. The bijectivity of the map A A allows to define the catenation of two partial arrays in a trivial way. b a b Example 5. The partial array A = a b is b b the companion of a partial array A of size (3, 3) where D(A) = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 1), (3, 3)} and H(A) = {(2, 1), (3, 2)}. a a 1n b b 1n X = and Y = a m1... a mn b m 1... b m n ijpam.eu

4 By column catenation we mean a a 1n X Y = a m1... a mn b b 1n b m 1... b m n By row catenation we mean a a 1n b b 1n X Y = a m1... a mn b m 1... b m n If A and B are two partial arrays of equal size then A is contained in B denoted by A B if D(A) D(B) and A(i, j) = B(i, j) for all (i, j) D(A) Definition 6. The partial arrays A and B are said to be compatible denoted by A B if there exists a partial array C such that A C and B C. By keeping the column fixed (8), a local R-period (Row period) of a partial array P with order (m, n) is a positive integer p such that P (i, j) = P (i + p, j) whenever (i, j) and (i + p, j) D(P ), 1 i m, 1 j n. In the same way by keeping the row fixed (8) a local C-period (Column period) of a partial array P with order (m, n) is a positive integer p such that P (i, j) = P (i, j + p) whenever (i, j) and (i, j + p) D(P ), 1 i m, 1 j n. Theorem 7. [7] Let P be a partial array of order (m, n) with one hole. If P is locally k R-periodic and locally l R-periodic and also if m k + l then P is gcd(k, l) R-periodic 3 Primitive Partial Arrays and its Fundamental Properties A partial array A is primitive if there exists no array B such that A B n with n 2 using row or column catenation accordingly. If B is primitive and B A then A is primitive. ijpam.eu

5 Example 8. ( ) a The partial array A = is primitive. b c Example 9. ( ) a The partial array A = is not primitive. b b Proposition 10. [3] Let u, v be non-empty words. Let y, z be partial words and let w be a word satisfying w u + v gcd( u, v ). If wy u m and wz v n for some integers m, n, then there exists a word x of length not greater than gcd( u, v ) such that u = x k and v = x l for some integers k, l. We now give proposition 10 s version of partial arrays. Proposition 11. Let A, B be non-empty arrays of order (i, j) & (i, k) respectively. Let Y, Z be partial arrays of order (i, m) and (i, n) respectively and let W be an array of order (i, l) satisfying l j + k gcd(j, k). If W Y A c and W Z V d for some integers c and d, then there exists an array X of order (i, r) with r gcd(j, k) such that A = X p and B = X q for some integers p and q using column catenation. Proof. Let W be the prefix of order (i, j + k gcd(j, k)) of W. Both j and k are periods of W. Then by Theorem 7 gcd(j, k) is also period of W. Hence there exists an array X of order (i, gcd(j, k)) such that W is contained in a power of X. ( ) ( ) a c e a c e a c e Example 12. A = B = ( ) ( b d f ) b d f b d f c e e a c e Y = Z = ( f d f ) d f a c e a c e a W = b d f b d f b then A = X 1 B = X 2 where X = and ( a c ) e b d f Note. In the above proposition i is fixed and so we are using column catenation. In the same way if j is fixed, we use row catenation. So proposition 11 is true for j fixed. ijpam.eu

6 Proposition 13. [3] Let u, v be non-empty words. Let y, z be partial words and let w be a partial word with one hole satisfying w u + v. If wy u m and wz (v) n for some integers m, n, then there exists a word x of length not greater than gcd( u, v ) such that u = x k and v = x l for some integers k, l. We now give extension of proposition 13 to partial arrays. Proposition 14. Let A, B be non-empty arrays of order (i, j) and (j, k) respectively. Let Y, Z be partial arrays of order (i, m) and (i, n) respectively and let W be a partial array of order (i, l) with at most one hole satisfying l j + k. If W Y A c and W Z B d for some integers c and d then there exists an array X of order (i, r) with r gcd(j, k) such that A = X p and B = X q for some integers p and q using column catenation. Proof. Let W be the prefix of order (i, j + k) of W. Both j and k are periods of W. By theorem 7 gcd(j, k) is also period of W. Hence there exists an array X of order (i, gcd(j, k)) such that W contained in a power of X. If H(W ) = φ then the result follows. If H(W ) = (p, q) where 1 p i and 1 q (j + k). If q gcd(j, k) then r = q and W (p, q + gcd(j, k)) = X(p, r). If q > gcd(j, k) then let r, 1 r < gcd(j, k) be the remainder of division of q by gcd(j, k). Then W (p, q gcd(j, k)) = X(p, r ) for all gcd(j, k). Since gcd(j, k) divides j and k, we conclude A = X p and B = X q for some integers p and q. Note. In the same way proposition 14 is true, if number of column is fixed. Theorem 15. [5] If u is a non-empty partial word, then there exists a primitive word v and a positive integer n such that u v n. For non-empty partial arrays the following result is true. Theorem 16. If A is non-empty partial array of order (i, j) then there exists a primitive array of order (i, k) and a positive integer n such that A B n using column catenation. ijpam.eu

7 Proof. Throughout the proof the number of rows is fixed and we use column catenation. We prove the result by induction on the number of columns (j) of A. For j = 1 the result is obvious. Assume that the result is true for number of columns less than j. Case (i). If a is primitive, assume B is any array of order (i, k) where k < j such that A B, then B is primitive and the result follows. Case (ii). If A is not primitive then A B n for some array B of order (i, k) and integer n 2. Since number of columns of B less than number of columns of A by induction hypothesis there exists a primitive array C and a positive integer m such that B C m. We ve A C mn. Proposition 17. [6] Let u be a partial word with one hole which is not of the form x x for any word x. If a and b are distinct letters, then ua or ub is primitive. The above result is true for partial arrays. Proposition 18. Let A be a partial array of order (i, j) with which is not of the form X A X for any array X of order (i, m) and A is the holes in array form of order (i, 1). If B and C are distinct arrays of order (i, 1) then AB or AC is primitive. Assume the AB V m, AC W n for some arrays V and W of order (i, k) and (i, l) respectively and of integers m 2, n 2. Now k and l periods of A and j = mk 1 = nl 1. Hence 2j = mk + nl 2 and j = mk + nl Since m, n 2 we get j k + l 1. Case (i) For j = k + l 1, k = l and m = n = 2. Since V ends with array B and W ends with array C. Put V = XB and W = Y C we get A XBX and A Y CY with X = Y. We conclude that A = X X, a contradiction. Case (ii) For j > k + l 1. By proposition 14 there exists an array X such that V = X P and W = X q for some integers p, q. Therefore AB X mp and AC X nq which is contradicting since B C. ijpam.eu

8 Proposition 19. Let A be a partial array with two holes. If AA XAY for some partial arrays X, Y implies X = ǫ or Y = ǫ, then A is primitive. Proof. Assume AA XAY for some partial arrays X, Y implies X = ǫ or Y = ǫ. Suppose A is not primitive. Then by definition there exists a partial array W and integer n 2 such that A W n. But AA W n 1 AW and using our assumption we get W n 1 = ǫ or W = ǫ, a contradiction. 4 Conclusion Motivated by primitivity on partial words we define primitivity on partial arrays and we verify the fundamental properties of primitivity. We prove for a partial array A there exist a primitive array B such that A B n for some positive integer n. References [1] Aldo de Luca, On the combinatorics of finite words, Theoretical Computer Science, 218 (1999), [2] J. Berstel, L. Boasson, Partial words and a theorem of Fine and Wilf, Theoretical Computer Science, 218 (1999), [3] F. Blanchet-Sadri, Primitive partial words, Discrete Applied Mathematics, 148 (2005), [4] F. Blanchet-Sadri, Arundhati R, Anavekar, Testing of primitive partial words, Discrete Applied Mathematics, 155 (2007), [5] F. Blanchet-Sadri, R.A. Hegstrom, Partial words and a theorem of Fine and Wilf revisited, Theoretical Computer Science, 270, No. 1/2 (2002), [6] F. Sweety, D.G. Thomas, T. Kalyani, Collage of Hexagonal Arrays, In G. Bebiset 91 editing International Symposium on Visual Computing Part - II, LNCS, 5359 (2008), ijpam.eu

9 [7] S. Vijayachitra, K. Sasikala, Periodicity in partial arrays,, 101, No. 7 (2015), ijpam.eu

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