A note on consecutive ones in a binary matrix

A note on consecutive ones in a binary matrix Lemnouar Noui Abstract. A binary matrix A is said to have the ”Consecutive Ones Property” (C1P) if its columns can be permuted so that in each row, the ones appear in one run (i.e., all ones are adjacent). The Consecutive Ones Submatrix (COS) problem is, given a binary matrix A and a positive integer m0 , to find m0 columns of A that form a submatrix with the C1P property. The matrix reordering problem is to find a matrix A′ obtained by permuting the columns of A that minimizes Cr (A) the number of sequences of consecutive ones in A. In this paper, by using two quadratic forms, we calculate the number Cr (A). We apply the obtained results to the orthogonal matrices and Hamming matrices, in addition, the two above problems can be solved for these matrices. M.S.C. 2000: 65F50, 05A05, 11H50, 11H71. Key words: consecutive ones, data compression, quadratic form, Hamming distance. §1. Introduction Let M be an n × m matrix over a field K. A run in a row of M is a sequence of non-zero entries. The cost runs of M is the sum Cr (M ) of the number of runs in each of its rows. In the notations of this paper, a binary matrix is a matrix with coefficients in R, but whose entries are only zeros or ones. A binary matrix M has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1’s consecutive in each row. The question of whether M has C1P was solved by Booth and Lueker [4] using a linear time algorithm. The problem of verifying the property C1P has applications in computational biology [3], recognizing interval graphs [5] and file organization [6, 9]. It is a special case of the two following problems. P1. Matrix Reordering. Given a binary matrix M find a matrix M ′ obtained by permuting the columns of M that minimizes Cr (M ′ ). This problem is NP-hard [1]. It has applications in data compression which is the process used to reduce the physical size of a block of information. P2. Consecutive Ones Submatrix (COS). Given a binary matrix M and a positive integer m0 , are there m0 columns of M that form a submatrix with C1P property? Applied Sciences, Vol.9, 2007, pp. 141-147. c Balkan Society of Geometers, Geometry Balkan Press 2007. ° 142 Lemnouar Noui Such a question has potential applications in physical mapping with hybridization data [2]. According to [7, 8] this problem is NP-complete. In his paper, by using two quadratic forms, we calculate the number Cr (A) of runs in a binary matrix A and we give a solution of the two above problems for the orthogonal matrices and Hamming matrices. §2. Main results 2.1 Computation of the number Cr (A) . Let E = Mk,n (K) be the vector space of k × n matrices over a field K and define the mapping φ : E × E → K by φ (A, B) = Tr (tBA). We denote by P the image of the identity matrix In from a permutation of its columns. Let us note by Ci the ith column and Rj the jth row of the matrix A and x · y = hx, yi the standard inner-product of two elements x and y of K n , then Proposition 1. a) φ is a symmetric nondegenerate bilinear form. b) If q is the quadratic form associated with φ, then ∀A ∈ E q (AP ) = q (A), that is, Sn the symmetric group of degree n is contained in the orthogonal group of q. c) If q ′ : E → K is the mapping defined by q ′ (A) = C1 C2 + · · · + Cn−1 Cn , then q ′ is a quadratic form (not necessarily nondegenerate). Proof. a) If A, A′ , B, B ′ are matrices in E, then ¡ ¢ ¡ ¢ ¡ ¢ φ (A + A′ , B) = Tr tB(A + A′ = Tr tBA + Tr tBA′ = φ (A, B) + φ (A′ , B) . Also, it is easy to see that φ (A, B + B ′ ) = φ (A, B) + φ (A, B ′ ) . For every λ ∈ K, we have φ (λA, B) = Tr (tB (λA)) = λφ (A, B) = φ (A, λB). Thus φ is a bilinear form and symmetric because Tr (tBA) = Tr (t (tBA)) = Tr (tAB). To prove that φ is nondegenerate, we assume that φ (A, B) = 0 for every matrix A in E. Let us denote by Ei,j the matrix whose entries equal zero except ei,j which equals 1. Let A = E1,1 and B = (bi,j ), so that Tr (tBA) = b1,1 = 0, in the general case, for A = Ei,j , we obtain bi,j = 0, therefore B = 0. b) The quadratic form q associated with φ is defined by q (A) = Tr (tA · A). Then q (A) = C12 + · · · + Cn2 = R12 + · · · + Rk2 and we conclude that φ is a positive form if K = R. Let P be a permutation matrix. Then ¡ ¢ ¡ ¡ ¢¢ q (AP ) = Tr t (AP ) AP = Tr tP tAAP . t Note that the matrices tP and tAAP are square and Tr[tP (tAAP )] = Tr[(tAAP ) P ]. Also P tP = In . Hence q (AP ) = q (A). Let fP : E → E be the linear mapping defined by fP (A) = AP . It is easy to verify that q (fP (A)) = q (A), that is, fP belongs to Oq (E) the orthogonal group of 143 A note on consecutive ones in a binary matrix q and Sn ⊂ Oq (E). c) Let ψ : E × E → K be the mapping defined by " # n−1 X 1 A1 B 2 + Ai (Bi−1 + Bi+1 ) + An Bn−1 , ψ (A, B) = 2 i=2 where Ai is the ith column of A and Bj is the jth column of B. It is easy to show that ψ is a symmetric bilinear form. This form is  not necessarily  nondegenerate. To see 1 0 this, it is enough to consider the matrix B =  0 0 1 0 for every matrix A in the space M3,3 (K). Obviously q ′ is the quadratic form associated with ψ. −1 0 , we have ψ (A, B) = 0 −1 Now, we shall compute the number of runs Cr (A) in the matrix A and minimize it. We have Proposition 2. a) q (A) is the number of ones in A. b) If q ′ is the quadratic form associated with ψ, then Cr (A) = q (A) − q ′ (A). c) Cr (A) is minimized if and only if q ′ (A) is maximized. Proof. a) Since Ci2 equals the number of ones in the ith column, q(A) = C12 + · · · + Cn2 is the number of ones in A. b) Let us write tA · A = (li,j ), then the number of runs in A is equal to Tr (tAA) − Pn−1 j=1 lj,j+1 , but we can express this by using the columns Ci , that is, Cr (A) = q (A) − q ′ (A) where q ′ (A) = C1 C2 + · · · + Cn−1 Cn . c) It follows from Proposition 1, that q (AP ) = q (A); thus q(A) is invariant by permuting the columns of A, then we deduce the last result. Remark 1. To simplify the computation of the number Cr (A), we distinguish the two cases: a) If n ≤ k, we write q(A) = C12 + · · · + Cn2 and q ′ (A) = C1 C2 + · · · + Cn−1 Cn . b) If k ≤ n, we write q(A) == R12 + · · · + Rk2 and q ′ (A) = q ′ (R1 ) + · · · + q ′ (Rk ) where Rm is the mth row of A. Example 1. Let the  1  0  A=  1  1 1 matrices 1 0 0 1 0 0 1 0 0 0 1 0 1 1 0       and   1 0 1 1 0 0 B =  0 1 1 0 1 0 · 1 1 1 0 0 0 Then for the first matrix, we have k = 5 and n = 4. Hence Cr (A) = q (A) − q ′ (A) = 10 − 2 = 8. 144 Lemnouar Noui For the second matrix, we have k = 3, n = 6 and q (B) = R12 + R22 + R32 = 9, q ′ (B) = q ′ (R1 ) + q ′ (R2 ) + q ′ (R3 ) = 4. So Cr (B) = 5. If x is a vector in K n , the Hamming weight wt (x) is the number of nonzero components in x, the Hamming distance d (x, y) between two vectors x and y is the Hamming weight of x − y. Let A = (ai,j ) be a matrix, let us note a′i,j = d (Ci , Cj ) where Ci is the ith column of A. Proposition 3. If A is an k × n binary matrix, then n−1 i 1 hX d (Ci , Ci+1 ) + wt (C1 ) + wt (Cn ) . Cr (A) = 2 i=1 2 Proof. Note that a′i,j = d (Ci , Cj ) = (Ci − Cj ) , by Proposition 2, C12 + Pn−1 i=1 d (Ci , Ci+1 ) + Cn2 Pn−1 2 = C¡12 + i=1 (Ci − Ci+1 ) + Cn2 ¢ = 2 C12 + · · · + Cn2 − C1 C2 − · · · − Cn−1 Cn = 2Nb (A) . Obviously Ci2 = wt (Ci ), consequently the last proposition is proved. 2.2 2.2.1 COS and Matrix Reordering problems Orthogonal matrices An n × n matrix M over the field K satisfying M tM = In is called an orthogonal matrix. Proposition 4. a) If A is a n × n binary orthogonal matrix, then Cr (A) = Cr (AP ) = n for every permutation matrix P. b)Every orthogonal binary matrix A has C1P property. Proof. a) Since A is orthogonal, q ′ (A) =0, by using Proposition 2, we have Cr (A) = q (A) = T r (A t A) = n. b) The orthogonal matrix A is invertible, then each row of A contains at least one non-zero element. By proposition 4 we have Cr (A) = n, so A has C1P property. For the first problem the suitable permutation for the orthogonal binary matrix is for example the identity permutation, as answer to the second problem m0 = n is the largest number of columns that form a submatrix with C1P property. A note on consecutive ones in a binary matrix 2.2.2 145 Hamming matrices ¡ ¢ Let n = 2k − 1 with k ≥ 2. The Hamming matrix is a k × 2k − 1 binary matrix whose of all nonzero binary vectors of length k:  columns consist  0  0  A=  .  0 1 0 0 . 1 0 0 0 . 1 1 . . . . . 1 1 . . 1   .   ¶ 1 2 ... n and Pk denote respectively a permutation of n = x1 x2 . . . xn 2k − 1 columns of A and the image of the identity matrix I2k −1 by this permutation. ¡ ¢ Proposition 5. Let A be an k × 2k − 1 Hamming matrix, then Let pk = µ a) The number of ones in A equals q (A) = k × 2k−1 , ¡ ¢ q ′ (Rm ) = 2m−1 × 2k−m − 1 and Cr (A) = 2k − 1. b) There is a permutation pk defined recursively as follows:  µ ¶ 1   p :  1    ¶ 1 µ  ′ 1 . m where m = 2k −1 − 1, then If pk′ −1 = x . x  1 m  ¶ µ   1 . m m+1 . 2m 2m + 1    pk ′ = x1 . xm xm + 2xm . x1 + 2xm 2xm such that Cr (APk ) is minimized, that is, Cr (APk ) = 2k−1 . Proof. a) If A is a Hamming matrix, its mth column is determined by writing the number m in the basic 2. Since every row of A consists of 2k−1 ones, it follows that q (A) = k × 2k−1 . As k ≤ n = 2k − 1, to find Cr (A) we write Cr (A) =¡ q (A) − q ′¢(A) = k · 2k−1 − (q ′ (R1 ) + q ′ (R2 ) + ... + q ′ (Rk )) where q ′ (Rm ) = 2m−1 × 2k−m − 1 , thus we obtain Cr (A) = 2k − 1. b) By proposition 3, in order to minimize Cr (AP ), it is enough to minimize d (Ci , Ci+1 ), wt (C1 ) and wt (Cn ) , with this intention we shall define a permutation pk recursively µ ¶ as follows 1 p1 : 1 ¶ µ 1 2 3 p2 : 1 3 2 : µ ¶ ′ 1 ... m pk′ −1 = with m = 2k −1 − 1 x1 ... xm ¶ µ 1 ... m m+1 ... 2m 2m + 1 pk ′ = · x1 ... xm xm + 2xm ... x1 + 2xm 2xm Then Cr (APk ) is minimized because d (Ci , Ci+1 ) = wt (C1 ) = wt (Cn ) = 1, so that £¡ ¢ ¤ Cr (APk ) = 21 2k − 1 − 1 + 2 = 2k−1 . ¡ ¢ Proposition 6. Let A be an k × 2k − 1 Hamming matrix, then a) A has C1P property if and only if k = 1 or k = 2. 146 Lemnouar Noui b) For k ′ ≤ 2k − 1 there exists an k ′ × k submatrix with C1P property. Proof. a) By proposition 5 Cr (APk ) = 2k−1 , then A has C1P property if and only if 2k−1 = k. The function f (x) = (x − 1)ln2 − lnx shows that 2k−1 = k ⇔ k = 1 or k = 2. b) If we take the 2k − 1 columns C2i −1 , 1 ≤ i ≤ k and C2k −2s , 1 ≤ s ≤ k − 1 of A, we obtain a submatrix A′ with C1P property. Now for k ′ ≤ 2k − 1 it is enough to take the first k ′ columns of A′ , so the COS problem can be solved for the Hamming matrices. The first problem is solved by proposition 5.   0 0 0 1 1 1 1 Example 2. Consider the 3 × 7 Hamming matrix A =  0 1 1 0 0 1 1 , 1 0 1 0 1 0 1 then k = 3, n = 23 − 1 and Cr (A) = 2k − 1 = 7. ¶ µ µ ¶ 1 2 3 1 , , p2 : To find the permutation p3 , we write p1 : 1 3 2 1 ¶ µ 1 2 3 4 5 6 7 , consequently, we have p3 : 1 3 2 6 7 5 4   0 0 0 1 1 1 1 AP3 =  0 1 1 1 1 0 0  and Cr (AP3 ) = 2k−1 = 4. 1 1 0 0 1 1 0 k ′ = 5is the largest number  of columns of A that form a submatrix 0 0 1 1 1 A′ =  0 1 1 1 0 with C1P property. 1 1 1 0 0 Remark 2. The traveling Salesman Problem (TSP) [1] requires that we find the shortest path visiting each of a given set of cities and returning to the starting point. The permutation pk gives a solution of the traveling Salesman problem for n = 2k − 1 cities     0 0 0 1  0  0  0  1       C1  C3   . C2  .  . ... Cn  .  0  1  1  1 1 0 1 1 for example if k = 3, it is easy to see that the distance is minimal for the path 1326754, see the figure below. 147 A note on consecutive ones in a binary matrix z ✻ C1 .. ...... ...... ¡ .... .... ¡................✍ ✛ ............ C5 ¡ ...... .. ...... ...... ..... . ...... ...... ...... ..... . ...... .... ¡ C4 ¡ ❄ ¡ ¡ x¡ ✠ ¡ ✲ C3 ¡ ¡ ¡C7 ✻ ❄ ¡ C2 ¡ ¡ ✠ C6 ✲ y References [1] D. Johnson, S. Krishnan, J.ChhuganI, S. Kumar and S. Venkatasubramanian, Compressing large Boolean matrices using reodering techniques, Proc. of the 30th VLDB Conference, Toronto, Canada, 2004, ???. [2] F. Alizadeh, R.M.Karp, D.K. and G. Zweig, Physical mapping of chromosomes using unique probes, Journal of Computational Biology 2 (1995), 159-1. [3] F. Annexstein, R. Swaminathan, On testing consecutive-ones property in parallel, Discrete Applied Mathematics 88 (1998), 7-28. [4] K.S. Booth, G. S. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. Comput. Systems Sci. 13 (3) (1976), 335-379. [5] J. S. Deogun, K. Gopalakrishnan, Consecutive retrieval property-revisited, Information Processing Letters 69 (1999), 15-20. [6] M. Flammini, G. Gambosi, S. Salomone, Boolean Routing, Lecture Notes in Computer Science 725, 1933. [7] M.R. Garey and D. S. Johnson, Computers and intractability: A guide to the theory of NP-completeness, San Francisco: W.H. Freeman, 1979. [8] S. P. Ghosh, File organization: the consecutive retrieval property, Commun. ACM 15 (1972), 802-808. [9] L. T. Kou, Polynomial complete consecutive information retrieval problems, SIAM J. Comput. 6 (1977), 67-75. Author’s address: Lemnouar Noui Department of mathematics, Faculty of Science, University of Batna, Algeria e-mail: [email protected]