Maximal Rectangles Research Articles

Regularity and the Gorenstein property of $L$-convex Polyominoes

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen–Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it. Though the main results are of algebraic nature, all proofs are combinatorial.

Indexing and querying color sets of images

We aim to study the set of color sets of continuous regions of an image given as a matrix of m rows over n≥m columns where each element in the matrix is an integer from [1,σ] named a color. The set of distinct colors in a region is called fingerprint. We aim to compute, index and query the fingerprints of all rectangular regions named rectangles. The set of all such fingerprints is denoted by F. A rectangle is maximal if it is not contained in a greater rectangle with the same fingerprint. The set of all locations of maximal rectangles is denoted by L. We first explain how to determine all the |L| maximal locations with their fingerprints in expected time O(nm2σ) using a Monte Carlo algorithm (with polynomially small probability of error) or within deterministic O(nm2σlog⁡(|L|nm2+2)) time. We then show how to build a data structure which occupies O(nmlog⁡n+|L|) space such that a query which asks for all the maximal locations with a given fingerprint f can be answered in time O(|f|+log⁡log⁡n+k), where k is the number of maximal locations with fingerprint f. If the query asks only for the presence of the fingerprint, then the space usage becomes O(nmlog⁡n+|F|) while the query time becomes O(|f|+log⁡log⁡n). We eventually consider the special case of squared regions (squares).

Complete branching rules for Specht modules

We give a combinatorial description for when the Specht module of an arbitrary diagram admits a (complete) branching rule. This description, given in terms of the maximal rectangles of the diagram, generalizes all previously known branching rules for Specht modules, such as those given by Reiner and Shimozono for northwest diagrams and by the present author for forest diagrams.

Computing Formal Concepts by Attribute Sorting

We present a novel approach to compute formal concepts of formal context. In terms of operations with Boolean matrices, the presented algorithm computes all maximal rectangles of the input Boolean matrix which are full of 1s. The algorithm combines basic ideas of previous approaches with our recent observations on the influence of attribute permutations and attribute sorting on the number of formal concepts which are computed multiple times. As a result, we present algorithm which computes formal concepts by successive context reduction and attribute sorting. We prove its soundness, discuss its complexity and efficiency, and show that it outperforms other algorithms from the CbO family in terms of substantially lower numbers of formal concepts which are computed multiple times.

Representing a Concept Lattice by Neighborly Ideas

Concept lattices are an ordering of the maximal rectangles dened by a binary relation. This paper, using the idea of neighbor sets in graph theory, presents an approach to obtain all the concepts for a context, and simultaneously gets the diagram of the concept lattice.

Partial vexillarity and bigrassmannian permutations

We define two closely related notions of degree for permutation patterns of type 2143. These give rise to classes of “ m-vexillary elements” in the symmetric group. Using partitions, the Ehresmann–Bruhat partial order, and sets constructed from permutation inversions, we characterize the m-vexillary elements. We relate the maximal bigrassmannian permutations in the (Ehresmann–Bruhat) order ideal generated by any given m-vexillary element w to the maximal rectangles contained in the shape of w.

Searching for complete set of free resource rectangles on FPGA area based on CPTR

As a coprocessor, field-programmable gate array (FPGA) is the hardware computing processor accelerating the computing capacity of computers. To efficiently manage the hardware free resources for the placing of tasks on FPGA and take full advantage of the partially reconfigurable units, good utilization of chip resources is an important and necessary work. In this paper, a new method is proposed to find the complete set of maximal free resource rectangles based on the cross point of edge lines of running tasks on FPGA area, and the prove process is provided to make sure the correctness of this method.

Sequenced spatiotemporal aggregation for coarse query granularities

Sequenced spatiotemporal aggregation (SSTA) is an important query for many applications of spatiotemporal databases, such as traffic analysis. Conceptually, an SSTA query returns one aggregate value for each individual spatiotemporal granule. While the data is typically recorded at a fine granularity, at query time a coarser granularity is common. This calls for efficient evaluation strategies that are granularity aware. In this paper, we formally define an SSTA operator that includes a data-to-query granularity conversion. Based on a discrete time model and a discrete 1.5 dimensional space model, we generalize the concept of time constant intervals to constant rectangles, which represent maximal rectangles in the spatiotemporal domain over which an aggregation result is constant. We propose an efficient evaluation algorithm for SSTA queries that takes advantage of a coarse query granularity. The algorithm is based on the plane sweep paradigm, and we propose a granularity aware event point schedule, termed gaEPS, and a granularity aware sweep line status, termed gaSLS. These data structures store space and time points from the input relation in a compressed form using a minimal set of counters. In extensive experiments, we show that for coarse query granularities gaEPS significantly outperforms a basic EPS that is based on an extension of previous work, both in terms of memory usage and runtime.

Representing a concept lattice by a graph

Concept lattices (also called Galois lattices) are an ordering of the maximal rectangles defined by a binary relation. In this paper, we present a new relationship between lattices and graphs: given a binary relation R, we define an underlying graph G R , and establish a one-to-one correspondence between the set of elements of the concept lattice of R and the set of minimal separators of G R . We explain how to use the properties of minimal separators to define a sublattice, decompose a binary relation, and generate the elements of the lattice.

Orthotreillis et séparabilité dans un graphe non orienté

We present a generalization of minimal separation in an undirected graph, and show how the maximal rectangles of the adjacency matrix describe such separators, and form an ortholattice, which we call the Separability Lattice. Furthermore, for any given ortholattice L, we show that there does not exist a unique minimal graph of which L is the Separability Lattice. We establish a necessary and sufficient condition for the existence of such a graph. The end of the paper deals with algorithmic considerations on the class of orthocom-plemented lattices.

A characterization of intersection graphs of the maximal rectangles of a polyomino

The interior of an orthogonal polygon drawn on a regular grid of the plane defines a set of cells (or squares) called a polyomino. We prove that the intersection graph of the maximal rectangles contained in a polyomino is slightly triangulated or has a star cutset.

Une note sur les contractions dans les polygones orthogonaux

We define a contraction for orthogonal polygons, and show that such a polygon can be transformed into a rectangle by a sequence of contractions. This result gives a new proof of the orthogonal art gallery theorem. We present also a property of the maximal rectangles contained in orthogonal polygons.

A new method of image compression using irreducible covers of maximal rectangles

The binary-image-compression problem is analyzed using irreducible cover of maximal rectangles. A bound on the minimum-rectangular-cover problem for image compression is given under certain conditions that previously have not been analyzed. It is demonstrated that for a simply connected image, the irreducible cover proposed uses less than four times the number of the rectangles in a minimum cover. With n pixels in a square, the parallel algorithm for obtaining the irreducible cover uses (n/log n) concurrent-read-exclusive write (CREW) processors in O(log n) time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stars, maximal rectangles, lattices: a new perspective on Q-analysis

A new star-hub structure of binary relations is discussed in the context of the methodology of Q-analysis, and parallels are drawn with maximal rectangles and Galois lattice structures. Although these structures generalize those of Q-analysis, there remain problems due to the very large number of star-hub pairs generated by fairly modest data sets. It is argued that more theory is necessary, and some possibilities are discussed. It is suggested that the criteria for defining new structures will come most fruitfully from the study of the relationship between backcloth and the ways it constrains traffic. Finally, it is argued that these combinatorial structures are still not sufficient to fully describe complex systems and that for this one needs to consider polyhedra in the context of N-ary relations.

Some Maximal Rectangles and the Realities of Applied Mathematics

Textbooks on elementary calculus tend to teach truth but not reality. The theory and methods handed us by Newton and Leibniz fill these tomes along with long lists of problems dealing with currently fashionable applications that teach little about the realities of problem solving. It is left to the instructor to provide the student with experiences that prompt intellectual growth and recognition of the major issues in applied mathematics. It is my hope that the examples discussed below will provide instructors with some ideas on easing students into the frame of mind required for solving the kind of problem that actually occurs. The device we will use here is the maximal rectangle idea. Specifically, let us look at the problem of determining the largest rectangle in the first quadrant with one side on the positive x-axis and inscribed under the graph of y = f(x). To simplify the details, assumef(O) = 0.

A Class of Perfect Graphs

Let P be a simply connected polyomino. Let $G( P )$ be the graph whose vertices are the maximal rectangles in P, two such vertices being adjacent if the corresponding rectangles have nontrivial intersection. In this paper we show that $G ( P )$ is perfect. This solves a problem posed by Berge et al.

Combinatorial properties of polyominoes

A finite set of cells in the infinite planar square grid is often called a polyomino. With each polyominoP, we may associate a hypergraph whose vertices are the cells ofP and whose edges are the maximal rectangles (in the standard position) contained inP. It turns out that these hypergraphs have many nice properties generalizing various properties of bipartite graphs and trees. We survey results in this direction.

Doppelt-einparametrische lineare optimierung.i:unabhängige parameter

In this paper are considered twofold-oneparametric linear programming problems with independing parameters, that means, linear programming problems, in which the coefficients of the objective function depend on one parameter, the components of the right sides of the constraints on another. The theoretical result consists in that the parameter plane, spanned by these two parameters is decomposed into maximal rectangles with edges parallel to axes so that for all pairs of parameters belonging to the same rectangle the same solution, still containing the right-side parameter is optimal and admissible resp. no solution exists. The rectangles, belonging to optimal admissible solutions are not overlapping. Having stated the theoretical result one describes a procedure for solving practically such problems and finaly there is given a nontrivial example.

Expansion of $k$-Schur functions for maximal rectangles within the affine nilCoxeter algebra

We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra.