Row Permutations Research Articles

Forbidden Configurations: Boundary Cases

Abstract A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say a (0,1)-matrix A has F ⊀ A , if no submatrix of A is a row and column permutation of F. Let the number of columns of A be ‖ A ‖ . Let forb ( m , F ) = max { ‖ A ‖ : A is m -rowed simple matrix and F ⊀ A } . A conjecture of Anstee and Sali predicts the asymptotics of forb ( m , F ) as a function of F. The conjecture identifies certain matrices F as boundary cases, namely k-rowed F with forb ( m , F ) = Θ ( m l ) while for any k × 1 column α that is not already present two or more times in F, we have forb ( m , [ F | α ] ) being Ω ( m l + 1 ) . We establish two new boundary cases and review two other newly identified boundary cases. This is further evidence for the conjecture.

Two-dimensional cache-oblivious sparse matrix–vector multiplication

In earlier work, we presented a one-dimensional cache-oblivious sparse matrix-vector (SpMV) multiplication scheme which has its roots in one-dimensional sparse matrix partitioning. Partitioning is often used in distributed-memory parallel computing for the SpMV multiplication, an important kernel in many applications. A logical extension is to move towards using a two-dimensional partitioning. In this paper, we present our research in this direction, extending the one-dimensional method for cache-oblivious SpMV multiplication to two dimensions, while still allowing only row and column permutations on the sparse input matrix. This extension requires a generalisation of the compressed row storage data structure to a block-based data structure, for which several variants are investigated. Experiments performed on three different architectures show further improvements of the two-dimensional method compared to the one-dimensional method, especially in those cases where the one-dimensional method already provided significant gains. The largest gain obtained by our new reordering is over a factor of 3 in SpMV speed, compared to the natural matrix ordering.

Forbidden configurations and repeated induction

For a given k×ℓ matrix F, we say a matrix A has no configurationF if no k×ℓ submatrix of A is a row and column permutation of F. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. We define forb(m,F) as the maximum number of columns in an m-rowed simple matrix which has no configuration F. A fundamental result of Sauer, Perles and Shelah, and Vapnik and Chervonenkis determines forb(m,Kk) exactly, where Kk denotes the k×2k simple matrix. We extend this in several ways. For two matrices G,H on the same number of rows, let [G∣H] denote the concatenation of G and H. Our first two sets of results are exact bounds that find some matrices B,C where forb(m,[Kk∣B])=forb(m,Kk) and forb(m,[Kk∣Kk∣C])=forb(m,[Kk∣Kk]). Our final result provides asymptotic boundary cases; namely matrices F for which forb(m,F) is O(mp) yet for any choice of column α not in F, we have forb(m,[F∣α]) is Ω(mp+1). This is evidence for a conjecture of Anstee and Sali. The proof techniques in this paper are dominated by repeated use of the standard induction employed in forbidden configurations. Analysis of base cases tends to dominate the arguments. For a k-rowed (0,1)-matrix F, we also consider a function req(m,F) which is the minimum number of columns in an m-rowed simple matrix for which each k-set of rows contains F as a configuration.

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.

Small forbidden configurations V: Exact bounds for 4 × 2 cases

The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to besimpleif it is a (0, 1)-matrix with no repeated columns. LetFbe ak× (0, 1)-matrix (the forbidden configuration). AssumeAis anm×nsimple matrix which has no submatrix which is a row and column permutation ofF. We define forb (m, F) as the largestn, which would depend onmandF, so that such anAexists.DefineFabcdas the (a+b+c+d) × 2 matrix consisting ofarows of [11],brows of [10],crows of [01] anddrows of [00]. With the exception ofF2110, we compute forb (m; Fabcd) for all 4 × 2Fabcd. A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb (m; F1111) = 4m− 4 and forb (m; F1210) = forb (m; F1201) = forb (m; F0310) = (2m)+m+ 2 (proofs are different). A different idea proves forb (m; F0220) = (2m) + 2m− 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb (m; F2110) is heavily related to designs and we offer some constructions of matrices avoidingF2110using existing designs.

Virtual Cylindrical View of a Color Image for its Permutation for an Encryption Purpose

This paper presents a novel algorithm for row and column permutation of pixels for the purpose of image encryption. The algorithm introduces a virtual cylinder surrounding an image and a virtual viewer looking at it but displaced from an original position. The key idea is based on the assumption that the light ray of each pixel to the viewer in her/his original position intersects the cylinder surface at a given point. When the viewer is displaced, the new position on a perpendicular image plane on which the pixel is projected should also have its direction intersecting the cylinder on the same point. As a result, all projected pixels in the new created image are slid from their original positions; but, some of them are delayed because they are piled up or projected out. In order to avoid information loss, these pixels are projected in the created holes of the new image. The consequence of such process is the creation of the expected permutation. Despite its simplicity, the algorithm shows a strong transformation of images for the purpose of their encryption.

Linear algebra methods for forbidden configurations

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m,F) as the maximum number of columns in a simple m-rowed matrix A for which no k×l submatrix of A is a row and column permutation of F. In set theory notation, F is a forbidden trace. For all k-rowed F (simple or non-simple) Furedi has shown that forb(m,F) is O(m k ). We are able to determine for which k-rowed F we have that forb(m,F) is O(m k−1) and for which k-rowed F we have that forb(m,F) is Θ(m k ). We need a bound for a particular choice of F. Define D 12 to be the k×(2 k −2 k−2−1) (0,1)-matrix consisting of all nonzero columns on k rows that do not have [11] in rows 1 and 2. Let 0 denote the column of k 0’s. Define F k (t) to be the concatenation of 0 with t+1 copies of D 12. We are able to show that forb(m,F k (t)) is Θ(m k−1). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Furedi and Sali. We provide a novel application of these methods. The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m,F).

Even and odd tournament matrices with minimum rank over finite fields

The (0, 1)-matrix A of order n is a tournament matrix provided A + A + I = J, where I is the identity matrix, and J = Jn is the all 1’s matrix of order n. It was shown by de Caen and Michael that the rank of a tournament matrix A of order n over a field of characteristic p satisfies rankp(A) ≥ (n − 1)/2 with equality if and only if n is odd and AA = O. This article shows that the rank of a tournament matrix A of even order n over a field of characteristic p satisfies rankp(A) ≥ n/2 with equality if and only if after simultaneous row and column permutations AA = [ ±Jm O O O ] , for a suitable integer m. The results and constructions for even order tournament matrices are related to and shed light on tournament matrices of odd order with minimum rank.

Two refinements of the bound of Sauer, Perles and Shelah, and of Vapnik and Chervonenkis

We say that a matrix is simple if it is a (0, 1)-matrix with no repeated columns. Given m and a k × l (0, 1)-matrix F , we seek the bound forb ( m , F ) on the maximum number of columns that a simple m -rowed matrix A can have if the simple matrix A has the property that it has no k × l submatrix which is a row and column permutation of F . Let K k denote the k × 2 k (0, 1)-matrix of all columns in k rows. Sauer, Perles and Shelah, and Vapnik and Chervonenkis established that forb ( m , K k ) is Θ ( m k − 1 ) . (An alternative description for the absence of a configuration K k is to say that the matrix has no shattered k -set.) We identify an easy but exact condition on a k -rowed simple matrix F for which forb ( m , F ) is O ( m k − 2 ) . A consequence is a classification of the asymptotics of forb ( m , F ) for simple four-rowed matrices F . In addition we identify an easy but exact description for all k -rowed non-simple matrices F which contain K k and for which forb ( m , F ) is still O ( m k − 1 ) . The results are further evidence for the conjecture of Anstee and Sali on the asymptotics of forb ( m , F ) for fixed F .

Hybrid Architecture and VLSI Implementation of the Cosine–Fourier–Haar Transforms

This paper presents a VLSI implementation of a novel hybrid architecture that computes three 8-point 1-D transforms—the discrete cosine transform, the discrete Fourier transform, and the Haar wavelet transform—on a single chip. The architecture is developed on matrix factorization and row permutation algorithms, where the basis forward transformation matrices are decomposed into common submatrices which are then shared among the transforms. A two-level hardware mapping has been employed which is parallel, pipelined, and multiplexed. The hybrid architecture, the first of its kind, is implemented using 0.18 μm CMOS technology. The estimated die size of the hybrid processor is 0.203 sq. mm, the frequency of operation is 100 MHz, the gate count is 20,400, and the power consumption is 15.38 mW. Compared to the existing designs, the proposed hybrid scheme has less power density and higher frequency of operation, making it very suitable for modern multimedia-based transcoding applications.

Extended substitution–diffusion based image cipher using chaotic standard map

This paper proposes an extended substitution–diffusion based image cipher using chaotic standard map [1] and linear feedback shift register to overcome the weakness of previous technique by adding nonlinearity. The first stage consists of row and column rotation and permutation which is controlled by the pseudo-random sequences which is generated by standard chaotic map and linear feedback shift register, second stage further diffusion and confusion is obtained in the horizontal and vertical pixels by mixing the properties of the horizontally and vertically adjacent pixels, respectively, with the help of chaotic standard map. The number of rounds in both stage are controlled by combination of pseudo-random sequence and original image. The performance is evaluated from various types of analysis such as entropy analysis, difference analysis, statistical analysis, key sensitivity analysis, key space analysis and speed analysis. The experimental results illustrate that performance of this is highly secured and fast.

Forbidden Configurations: Exact Bounds Determined by Critical Substructures

We consider the following extremal set theory problem. Define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. An $m$-rowed simple matrix corresponds to a family of subsets of $\{1,2,\ldots ,m\}$. Let $m$ be a given integer and $F$ be a given (0,1)-matrix (not necessarily simple). We say a matrix $A$ has $F$ as a configuration if a submatrix of $A$ is a row and column permutation of $F$. We define $\hbox{forb}(m,F)$ as the maximum number of columns that a simple $m$-rowed matrix $A$ can have subject to the condition that $A$ has no configuration $F$. We compute exact values for $\hbox{forb}(m,F)$ for some choices of $F$ and in doing so handle all $3\times 3$ and some $k\times 2$ (0,1)-matrices $F$. Often $\hbox{forb}(m,F)$ is determined by $\hbox{forb}(m,F')$ for some configuration $F'$ contained in $F$ and in that situation, with $F'$ being minimal, we call $F'$ a critical substructure.

Some new results for system decoupling and pole assignment problems

In a related article, we derived a canonical decomposition of the right invertible system { C , A , B } and applied this canonical decomposition to study the Smith form of the matrix pencil P ( s ) = ( A − s I B C 0 ) and findout the finite zeros and infinite zeros of P ( s ) , the range of the ranks of P ( s ) for s ∈ C , and the controllability of the right invertible system. In this paper, we will apply this canonical decomposition of the right invertible system { C , A , B } to deduce the triangular decouple upon to row permutation, provide some new results of the row-by-row decoupling, and associated pole assignment problems.

Variable neighbourhood search for bandwidth reduction

The problem of reducing the bandwidth of a matrix consists of finding a permutation of rows and columns of a given matrix which keeps the non-zero elements in a band as close as possible to the main diagonal. This NP-complete problem can also be formulated as a vertex labelling problem on a graph, where each edge represents a non-zero element of the matrix. We propose a variable neighbourhood search based heuristic for reducing the bandwidth of a matrix which successfully combines several recent ideas from the literature. Empirical results for an often used collection of 113 benchmark instances indicate that the proposed heuristic compares favourably to all previous methods. Moreover, with our approach, we improve best solutions in 50% of instances of large benchmark tests.

Group-theoretic analysis of cayley-graph-based cycle gf(2p) codes

Using group theory, we analyze cycle GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> ) codes that use Cayley graphs as their associated graphs. First, we show that through row and column permutations the parity check matrix H can be put in a concatenation form of row-permuted block-diagonal matrices. Encoding utilizing this form can be performed in linear time and in parallel. Second, we derive a rule to determine the nonzero entries of H and present determinate and semi-determinate codes. Our simulations show that the determinate and semi-determinate codes have better performance than codes with randomly generated nonzero entries for GF(16) and GF(64), and have similar performance for GF(256). The constructed determinate and semi-determinate codes over GF(64) and GF(256) can outperform the binary irregular counterparts of the same block lengths. One distinct advantage for determinate and semi-determinate codes is that they greatly reduce the storage cost of H for decoding. The results in this correspondence are appealing for the implementation of efficient encoders and decoders for this class of promising LDPC codes, especially when the block length is large.

High-Throughput Layered LDPC Decoding Architecture

This paper presents a high-throughput decoder architecture for generic quasi-cyclic low-density parity-check (QC-LDPC) codes. Various optimizations are employed to increase the clock speed. A row permutation scheme is proposed to significantly simplify the implementation of the shuffle network in LDPC decoder. An approximate layered decoding approach is explored to reduce the critical path of the layered LDPC decoder. The computation core is further optimized to reduce the computation delay. It is estimated that 4.7 Gb/s decoding throughput can be achieved at 15 iterations using the current technology.

Sufficient conditions for permutation equivalence to a WHS-matrix

Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins–Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A −1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix.

A diagonal measure and a local distance matrix to display relations between objects and variables

Proper permutation of data matrix rows and columns may result in plots showing striking information on the objects and variables under investigation. To control the permutation first, a diagonal matrix measureD was defined expressing the size relations of the matrix elements. D is essentially the absolute norm of a matrix where the matrix elements are weighted by their distance to the matrix diagonal. Changing the order of rows and columns increases or decreases D. Monte Carlo technique was used to achieve maximum D in the case of the object distance matrix or even minimal D in the case of the variable correlation matrix to get similar objects or variables close together. Secondly, a local distance matrix was defined, where an element reflects the distances of neighboring objects in a limited subspace of the variables. Due to the maximization of D in the local distance matrix by row and column changes of the original data matrix, the similar objects were arranged close to each other and simultaneously the variables responsible for their similarity were collected close to the diagonal part defined by these objects. This combination of the diagonal measure and the local distance matrix seems to be an efficient tool in the exploration of hidden similarities of a data matrix. Copyright © 2009 John Wiley & Sons, Ltd.

The simultaneous consecutive ones problem

The standard consecutive ones problem is concerned with permuting the columns of a 0/1-matrix in such a way that in every row all 1-entries occur consecutively. In this paper we study this problem with the additional requirement that also in every column the 1-entries have to be consecutive. To achieve this column permutations have to be allowed as well. We show that the weighted simultaneous consecutive ones problem is NP-hard and consider two special cases with fixed row and column permutations where one is still NP-hard and the other one turns out to be easy.

On the construction of low-density parity-check codes with girth 10

In this letter, construction of a class of ( 3 , k ) LDPC codes is presented. The proposed codes are constructed in 2-D lattices; by selection of slope-sets in the lattices, general triangles and quadrangles are removed, which leads to the corresponding codes to have girth no less than 10. By column and row permutation, the check matrix could be introduced to quasi-cyclic structure, which leads to encoding efficient, and the experiment shows that the codes have large minimum distance, and the bit error rate performance over AWGN channels is excellent.