Axis-parallel Rectangles Research Articles

4-labelings and grid embeddings of plane quadrangulations

A straight-line drawing of a planar graph G is a closed rectangle-of-influence drawing if for each edge uv, the closed axis-parallel rectangle with opposite corners u and v contains no other vertices. We show that each quadrangulation on n vertices has a closed rectangle-of-influence drawing on the (n−3)×(n−3) grid.The algorithm is based on angle labeling and simple face counting in regions. This answers the question of what would be a grid embedding of quadrangulations analogous to Schnyder’s classical algorithm for embedding triangulations and extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden.A further compaction step yields a straight-line drawing of a quadrangulation on the (⌈n2⌉−1)×(⌈3n4⌉−1) grid. The advantage over other existing algorithms is that it is not necessary to add edges to the quadrangulation to make it 4-connected.

Coloring half-planes and bottomless rectangles

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover, we give efficient coloring algorithms.

Locating an axis-parallel rectangle on a Manhattan plane

In this paper we consider the problem of locating an axis-parallel rectangle in the plane such that the sum of distances between the rectangle and a finite point set is minimized, where the distance is measured by the Manhattan norm l 1. In this way we solve an extension of the Weber problem to extensive facility location. As a model, our problem is appropriate for position sensing of rectangular objects.

A Doubly Exponentially Crumbled Cake

Abstract We consider the following cake cutting game: Alice chooses a set P of n points in the square (cake) [ 0 , 1 ] 2 , where ( 0 , 0 ) ∈ P ; Bob cuts out n axis-parallel rectangles with disjoint interiors, each of them having a point of P as the lower left corner; Alice keeps the rest. It has been conjectured that Bob can always secure at least half of the cake. This remains unsettled, and it is not even known whether Bob can get any positive fraction independent of n. We prove that if Alice can force Bobʼs share to tend to zero, then she must use very many points; namely, to prevent Bob from gaining more than 1 / r of the cake, she needs at least 2 2 Ω ( r ) points.

On sum coloring and sum multi-coloring for restricted families of graphs

We consider the sum coloring (chromatic sum) problem and the sum multi-coloring problem for restricted families of graphs. In particular, we consider the graph classes of proper intersection graphs of axis-parallel rectangles, proper interval graphs, and unit disk graphs. All the above-mentioned graph classes belong to a more general graph class of ( k + 1 ) -clawfree graphs (respectively, for k = 4 , 2 , 5 ). We prove that sum coloring is NP-hard for penny graphs and unit square graphs which implies NP-hardness for unit disk graphs and proper intersection graphs of axis-parallel rectangles. We show a 2-approximation algorithm for unit square graphs, with the assumption that the geometric representation of the graph is given. For sum multi-coloring, we confirm that the greedy first-fit coloring, after ordering vertices by their demands, achieves a k -approximation for the preemptive version of sum multi-coloring on ( k + 1 ) -clawfree graphs. Finally, we study priority algorithms as a model for greedy algorithms for the sum coloring problem and the sum multi-coloring problem. We show various inapproximation results under several natural input representations.

Directional discrepancy in two dimensions

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.

Minimum vertex cover in rectangle graphs

We consider the Minimum Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R 1 ∖ R 2 is connected for every pair of rectangles R 1 , R 2 ∈ R . This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of ( 1.5 + ε ) in general rectangle families, for any fixed ε > 0 , and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a novel way.

Evolutionary selection of hyperrectangles in nested generalized exemplar learning

The nested generalized exemplar theory accomplishes learning by storing objects in Euclidean n-space, as hyperrectangles. Classification of new data is performed by computing their distance to the nearest “generalized exemplar” or hyperrectangle. This learning method allows the combination of the distance-based classification with the axis-parallel rectangle representation employed in most of the rule-learning systems. In this paper, we propose the use of evolutionary algorithms to select the most influential hyperrectangles to obtain accurate and simple models in classification tasks. The proposal has been compared with the most representative models based on hyperrectangle learning; such as the BNGE, RISE, INNER, and SIA genetics based learning approach. Our approach is also very competitive with respect to classical rule induction algorithms such as C4.5Rules and RIPPER. The results have been contrasted through non-parametric statistical tests over multiple data sets and they indicate that our approach outperforms them in terms of accuracy requiring a lower number of hyperrectangles to be stored, thus obtaining simpler models than previous NGE approaches. Larger data sets have also been tackled with promising outcomes.

The Parameterized Complexity of Stabbing Rectangles

The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set R of axis-parallel rectangles in the plane, a set L of horizontal and vertical lines in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time within a factor of two, its parameterized complexity with respect to the parameter k was open so far. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k, which in particular means that there is no hope for an algorithm running in f(k)⋅|R∪L| O(1) time. Our reductions also show the W[1]-completeness of the more general problem Set Cover on instances that “almost have the consecutive-ones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row. We also show that the special case of Rectangle Stabbing where all rectangles are squares of the same size is W[1]-hard. The case where the input consists of non-overlapping rectangles was open for some time and has recently been shown to be fixed-parameter tractable (Heggernes et al., Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing, 2009). By giving an algorithm running in (2k) k ⋅|R∪L| O(1) time, we show that Rectangle Stabbing is fixed-parameter tractable in the still NP-hard case where both these restrictions apply, that is, in the case of disjoint squares of the same size. This algorithm is faster than the one in Heggernes et al. (Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing, 2009) for the disjoint rectangles case. Moreover, we show fixed-parameter tractability for the restrictions where the rectangles have bounded width or height or where each horizontal line intersects only a bounded number of rectangles.

Straight-Line Rectangular Drawings of Clustered Graphs

We show that every c-planar clustered graph has a straight-line c-planar drawing in which each cluster is represented by an axis-parallel rectangle, thus solving a problem posed by Eades, Feng, Lin, and Nagamochi (Algorithmica 44(1):1–32, 2006).

Closed rectangle-of-influence drawings for irreducible triangulations

A ( weak) closed rectangle-of-influence (RI for short) drawing is a straight-line planar grid drawing in which there is no other vertex inside or on the boundary of the axis parallel rectangle defined by the two end vertices of any edge. Biedl et al. (1999) [1] showed that a plane graph G has a closed RI drawing, if and only if it has no filled 3 -cycle (a cycle of 3 vertices such that there is a vertex in the proper interior). They also showed that such a graph G has a closed RI drawing in an ( n − 1 ) × ( n − 1 ) grid, where n is the number of vertices in G. They raised an open question on whether this grid size bound can be improved (Biedl et al., 1999 [1]). Without loss of generality, we investigate maximal plane graphs admitting closed RI drawings in this paper. They are plane graphs with a quadrangular exterior face, triangular interior faces and no filled 3-cycles, known as irreducible triangulations (Fusy, 2009 [2]). In this paper, we present a linear time algorithm that computes closed RI drawings for irreducible triangulations. Given an arbitrary irreducible triangulation G with n vertices, our algorithm produces a closed RI drawing with size at most ( n − 3 ) × ( n − 3 ) ; and for a random irreducible triangulation, the expected grid size of the drawing is ( 22 n 27 + O ( n ) ) × ( 22 n 27 + O ( n ) ) . We then prove that for arbitrary n ⩾ 4 , there is an n-vertex irreducible triangulation, such that any of its closed RI drawing requires a grid of size ( n − 3 ) × ( n − 3 ) . Thus the grid size of the drawing produced by our algorithm is tight. This lower bound also answers the open question posed in Biedl et al. (1999) [1] negatively.

Indecomposable Coverings

AbstractWe prove that for every k > 1, there exist k-fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every k > 1 a set of points P and a family of disks in the plane, each containing at least k elements of P, such that, no matter how we color the points of P with two colors, there exists a disk D ∈ all of whose points are of the same color.

Coloring axis-parallel rectangles

For every k and r, we construct a finite family of axis-parallel rectangles in the plane such that no matter how we color them with k colors, there exists a point covered by precisely r members of the family, all of which have the same color. For r = 2 , this answers a question of S. Smorodinsky [S. Smorodinsky, On the chromatic number of some geometric hypergraphs, SIAM J. Discrete Math. 21 (2007) 676–687].

Online conflict-free coloring for halfplanes, congruent disks, and axis-parallel rectangles

We present randomized algorithms for online conflict-free coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearly-equal axis-parallel rectangles. In all three cases, the coloring algorithms use O (log n ) colors, with high probability. We also present a deterministic algorithm for online CF coloring of points in the plane with respect to nearly-equal axis-parallel rectangles, using O (log 3 n ) colors. This is the first efficient (i.e, using polylog( n ) colors) deterministic online CF coloring algorithm for this problem.

A new fast heuristic for labeling points

In the point site labeling problem, we are given a set P = { p 1 , p 2 , … , p n } of point sites in the plane. The label of a point p i is an axis-parallel rectangle of specified size. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. Here, we investigate a special class of the point site labeling problem where (i) height of the labels of all the points are same but their lengths may differ, (ii) the label of a point p i touches the point at one of its four corners, and (iii) the label of one point does not obscure any other point in P. We describe an efficient heuristic algorithm for this problem which runs in O ( n n ) time in the worst case. We run our algorithm as well as the algorithm Rules proposed by Wagner et al. on randomly generated point sets and on the available benchmarks. The results produced by our algorithm are almost the same as Rules in most of the cases. But our algorithm is faster than Rules in dense instance. We have also computed the optimum solutions for all the examples we have considered by designing an algorithm, which performs an exhaustive search in the worst case. We found that the exhaustive search algorithm runs reasonably fast for most of the examples we have considered.

Delaunay graphs of point sets in the plane with respect to axis‐parallel rectangles

AbstractGiven a point set P in the plane, the Delaunay graph with respect to axis‐parallel rectangles is a graph defined on the vertex set P, whose two points p,q ∈ P are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94–136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(nlog 2 log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn log log n/(log n log log log n).We give two further applications of our methods: (1) We construct two‐dimensional n‐element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473–475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41–48). (2) For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis‐parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer‐Verlag, New York, 2005). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Dynamic Connectivity for Axis-Parallel Rectangles

In this paper we give a fully dynamic data structure to maintain the connectivity of the intersection graph of n axis-parallel rectangles. The amortized update time (insertion and deletion of rectangles) is and the query time (deciding whether two given rectangles are connected) is O(1). It slightly improves the update time (O(n 0.94)) of the previous method while drastically reducing the query time (near O(n 1/3)). In addition, our method does not use fast matrix multiplication results and supports a wider range of queries.

Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not necessarily a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n 1.5/log n)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n−1)×(n−1) integer grid, where n is the number of vertices in G.

Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs

In the rectangle stabbing problem, we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this article, we study the capacitated version of this problem in which the input includes an integral capacity for each line. The capacity of a line bounds the number of rectangles that the line can cover. We consider two versions of this problem. In the first, one is allowed to use only a single copy of each line ( hard capacities ), and in the second, one is allowed to use multiple copies of every line, but the multiplicities are counted in the size (or weight) of the solution ( soft capacities ). We present an exact polynomial-time algorithm for the weighted one dimensional case with hard capacities that can be extended to the one dimensional weighted case with soft capacities. This algorithm is also extended to solve a certain capacitated multi-item lot-sizing inventory problem with joint set-up costs. For the case of d -dimensional rectangle stabbing with soft capacities, we present a 3 d -approximation algorithm for the unweighted case. For d -dimensional rectangle stabbing problem with hard capacities, we present a bi-criteria algorithm that computes 4 d -approximate solutions that use at most two copies of every line. Finally, we present hardness results for rectangle stabbing when the dimension is part of the input and for a two-dimensional weighted version with hard capacities.

On the Algorithmic Aspects of Discrete and Lexicographic Helly-Type Theorems and the Discrete LP-Type Model

Helly's theorem says that, if every $d+1$ elements of a given finite set of convex objects in $\mathbb{R}^d$ have a common point, there is a point common to all of the objects in the set. In discrete Helly theorems the common point should belong to an a priori given set. In lexicographic Helly theorems the common point should not be lexicographically greater than a given point. Using discrete and lexicographic Helly theorems we get linear time solutions for various optimization problems. For this, we introduce the DLP-type (discrete linear programming-type) model, and provide new algorithms that solve in randomized linear time fixed-dimensional DLP-type problems. For variable-dimensional DLP-type problems, our algorithms run in time subexponential in the combinatorial dimension. Finally, we use our results in order to solve in randomized linear time problems such as the discrete $p$-center on the real line, the discrete weighted 1-center problem in $\mathbb{R}^d$ with either $l_1$ or $l_\infty$ norm, the standard (continuous) problem of finding a line transversal for a totally separable set of planar convex objects, a discrete version of the problem of finding a line transversal for a set of axis-parallel planar rectangles, and the (planar) lexicographic rectilinear $p$-center problem for $p=1,2,3$. These are the first known linear time algorithms for these problems. Moreover, we use our algorithms to solve in randomized subexponential time various problems in game theory, improving upon the best known algorithms for these problems.