A Note on Deterministic Poly-Time Algorithms for Partition Functions Associated with Boolean Matrices with Prescribed Row and Column Sums

We consider the problem of counting the number of contingency tables with given row and column sums. This problem is known to be #P-complete, even when there are only two rows [7]. In this paper we present the first fully-polynomial randomized approximation scheme for counting contingency tables when the number of rows is constant. A novel feature of our algorithm is that it is a hybrid of an exact counting technique with an approximation algorithm, giving two distinct phases. In the first, the columns are partitioned into "small" and "large". We show that the number of contingency tables can be expressed as the weighted sum of a polynomial number of new instances of the problem, where each instance consists of some new row sums and the original large column sums. In the second phase, we show how to approximately count contingency tables when all the column sums are large. In this case, we show that the solution lies in approximating the volume of a single convex body, a problem which is known to be solvable in polynomial time [5].

The maximum absolute value of the determinant of n x n nonsinguiar (0,1) matrices that have constant line sums (i.e., row sums and column sums) k is investigated. For n≠4k=2, this maximum determinant is determined to be 2t if n=3t or 3t+2, and 2t−1 if n=3t+1. Restriction to a subset of these matrices, namely those that are symmetric and have zero trace (their graphs are regular of degree k), leaves the maximum unchanged for k=2. For this restricted class, when n≥7k=n−3, the maximum absolute value of the determinant is (n−3)3[n/4]−-1, This maximum gives a lower bound for the maximum absolute value of the determinant for the larger class, but in general this bound is not tight. Other determinantal values and bounds for specific n and k are given.

Asymptotic enumeration of sparse 0–1 matrices with irregular row and column sums

where the biological activity is expressed as the reciprocal of the concentration that elicit a given biological, pharmacological or toxicological endpoint such as ED 50 (effective dose 50%), IC 50 (inhibitory concentration 50%) or LD 50 (lethal dose 50%), and where x i is the i predictor (from a set of n predictors) raised to the powers from 1 to m and weighted by its coefficients a ij obtained from a multivariate statistical fitting [4,5]. Usually, most of the coefficients a ij can be zero, leading to much simpler looking QSAR models than the general form in Equation 1, as in the example below. Often, other functions of x i are used in the QSAR model such as the logarithmic or exponential function instead of a simple power expansion as in Equation 1. As a simple illustration of the operation of Equation 1 is the following example of a QSAR model to predict the inhibitory effect of 3-nitroflavones (3NF) based on a single predictor, x (the energy of the highest occupied molecular orbital, E HOMO ) raised to the power 1 (see also [6]). Invasive tumor growth and metastasis are sustained by angiogenesis (the formation of new blood vessels [7,8]), which led Folkman to suggest anticancer therapies by the use of antiangiogenic agents [9,10]. Lichtenberg et al. have shown that the antiangiogenicity of six 3NF is correlated with the energies of the highest occupied molecular orbital, E HOMO, the lower the oxidation potential the greater the inhibitory effects on several biomarkers of angiogenesis [11,12]. These authors report the following correlation between the E HOMO and the IC 50 of six 3NF for inhibition of proliferation of microvascular endothelial HMEC-1 cells after 72 h of incubation:

We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts.The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ2 n), where λ is the edge connectivity, but λ could be Ω(n).At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with ~O(m/δ) edges which preserves all minimum cuts of G with at least two vertices on each side.In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.

On (0, 1)-matrices with prescribed row and column sum vectors

The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. Our deterministic algorithm is almost as fast as the randomized algorithms mentioned above.

The network flows approach for matrices with given row and column sums

General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices, the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relates to the pseudoinverses of its component parts. For invertible matrices, this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.

This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri (i = 1, … , m) and let the sum of column i of A be denoted by si (i = 1, … , n).

Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in Õ(kmn \(^{\rm 1/{\it k}}\)) expected time and construct a data structure (a (2k –1)-approximate distance oracle) of size O(kn \(^{\rm 1+1/{\it k}}\)) capable of returning in O(k) time an approximation \(\hat{\delta}(u,v)\) of the distance δ(u,v) from u to v in G that satisfies \(\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)\), for any two vertices u,v∈ V. They also presented a much slower Õ(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn \(^{\rm 1+1/{\it k}}\)log n). We present here a deterministic Õ(kmn \(^{\rm 1/{\it k}}\)) time algorithm for constructing oracles of size O(kn \(^{\rm 1+1/{\it k}}\)). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor.Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn \(^{\rm 1+1/{\it k}}\)) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.

Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer $r$, all complete row and column sums are $r$, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin $r/2$ statistical mechanical vertex models with domain-wall boundary conditions. The case $r=1$ gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size $n$ are the integer points of the $r$-th dilate of an integral convex polytope of dimension $(n{-}1)^2$ whose vertices are the standard alternating sign matrices of size $n$. It then follows that, for fixed $n$, these matrices are enumerated by an Ehrhart polynomial in $r$.

Let be real n-tuplesn>3, satisfying It is shown that a necessary and sufficient condition for the existence of a non-negative matrix with main diagonal with row sums and column sums is that ft Equality can hold if and only if all the off-diagonal positive entries of the matrix are restricted to the kth row and the fcth column, for some k,1<k<n.

Inn then last two decades the Content Based Image Retrieval (CBIR) considered as one of the topic of interest for theresearchers. It depending one analysis of the image’s visual content which can be done by extracting the color, texture and shapefeatures. Therefore, feature extraction is one of the important steps in CBIR system for representing the image completely. Color featureis the most widely used and more reliable feature among the image visual features. This paper reviews different methods, namely LocalColor Histogram, Color Correlogram, Row sum and Column sum and Colors Coherences Vectors were used to extract colors featurestaking in consideration the spatial information of the image.