Axis-parallel Boxes Research Articles

An improved algorithm for Kleeʼs measure problem on fat boxes

The measure problem of Klee asks for the volume of the union of n axis-parallel boxes in a fixed dimension d. We give an O(n(d+2)/3) time algorithm for the special case of all boxes being cubes or, more generally, fat boxes. Previously, the fastest run-time was nd/22O(log⁎n), achieved by the general case algorithm of Chan [SoCG 2008]. For the general problem our run-time would imply a breakthrough for the k-clique problem.

Quick detection of brain tumors and edemas: A bounding box method using symmetry

A significant medical informatics task is indexing patient databases according to size, location, and other characteristics of brain tumors and edemas, possibly based on magnetic resonance (MR) imagery. This requires segmenting tumors and edemas within images from different MR modalities. To date, automated brain tumor or edema segmentation from MR modalities remains a challenging, computationally intensive task. In this paper, we propose a novel automated, fast, and approximate segmentation technique. The input is a patient study consisting of a set of MR slices, and its output is a subset of the slices that include axis-parallel boxes that circumscribe the tumors. Our approach is based on an unsupervised change detection method that searches for the most dissimilar region (axis-parallel bounding boxes) between the left and the right halves of a brain in an axial view MR slice. This change detection process uses a novel score function based on Bhattacharya coefficient computed with gray level intensity histograms. We prove that this score function admits a very fast (linear in image height and width) search to locate the bounding box. The average dice coefficients for localizing brain tumors and edemas, over ten patient studies, are 0.57 and 0.52, respectively, which significantly exceeds the scores for two other competitive region-based bounding box techniques.

Piercing random boxes

Consider a set of n random axis parallel boxes in the unit hypercube in ${\bf R}^{d}$ **image** , where d is fixed and n tends to infinity. We show that the minimum number of points one needs to pierce all these boxes is, with high probability, at least $\Omega_d(\sqrt{n}(\log n)^{d/2-1})$ **image** and at most $O_d(\sqrt{n}(\log n)^{d/2-1}\log \log n)$ **image** . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 365–380, 2011 © 2011 Wiley Periodicals, Inc.

Box-respecting colorings of [formula omitted]-dimensional guillotine-partitions

A strong box-respecting coloring of an n -dimensional box-partition is a coloring of the vertices of its boxes with 2 n colors such that any box has all the colors appearing on its 2 n vertices. This is a generalization of rectangle-respecting colorings and strong polychromatic colorings to more than two dimensions. A guillotine-partition is obtained by starting with a single axis-parallel box and recursively cutting a box of the partition into two boxes by a hyperplane orthogonal to one of the n coordinate axes. We prove that there is a strong box-respecting coloring of any n -dimensional guillotine-partition. This theorem generalizes the result of Horev et al. (2009) [7] who proved the two-dimensional case. The proof gives an efficient coloring algorithm as well.

Boxicity of Leaf Powers

The boxicity of a graph G, denoted as boxi(G), is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are used in the construction of phylogenetic trees in evolutionary biology and have been studied in many recent papers. We show that for a k-leaf power G, boxi(G) ≤ k−1. We also show the tightness of this bound by constructing a k-leaf power with boxicity equal to k−1. This result implies that there exist strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.

A note about weak ε-nets for axis-parallel boxes in d-space

We show the existence of weak ε-nets of size O ( 1 / ε log log ( 1 / ε ) ) for point sets and axis-parallel boxes in R d , for d ⩾ 4 . Our analysis uses a non-trivial variant of the recent technique of Aronov et al. (2009) [3] that yields (strong) ε-nets, whose size have the above asymptotic bound, for d = 2 , 3 .

Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes

We show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$ for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in $\boldsymbol{R}^3$ and axis-parallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of $\varepsilon$-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\log\frac{1}{\varepsilon}\right)$ for the dual range space of “fat” regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the hitting set or the set cover problems associated with the corresponding range spaces.

A (slightly) faster algorithm for Klee's measure problem

Given n axis-parallel boxes in a fixed dimension d ⩾ 3 , how efficiently can we compute the volume of the union? This standard problem in computational geometry, commonly referred to as Klee's measure problem, can be solved in time O ( n d / 2 log n ) by an algorithm of Overmars and Yap (FOCS 1988). We give the first (albeit small) improvement: our new algorithm runs in time n d / 2 2 O ( log ∗ n ) , where log ∗ denotes the iterated logarithm. For the related problem of computing the depth in an arrangement of n boxes, we further improve the time bound to near O ( n d / 2 / log d / 2 − 1 n ) , ignoring log log n factors. Other applications and lower-bound possibilities are discussed. The ideas behind the improved algorithms are simple.

Construction of Minimal Bracketing Covers for Rectangles

We construct explicit $\delta$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $\delta$-bracketing covers are bounded from above by $\delta^{-2} + o(\delta^{-2})$. A lower bound for the cardinality of arbitrary $\delta$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $\delta^{-2}+O(\delta^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $\delta$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $\delta^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $\delta$.

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R 1×R 2×⋅⋅⋅×R k where R i (for 1≤i≤k) is a closed interval of the form [a i ,b i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}$approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(Δ+2)ln n⌉ dimensions, where Δ is the maximum degree of G. This algorithm implies that box (G)≤⌈(Δ+2)ln n⌉ for any graph G. Our bound is tight up to a factor of ln n. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree Δ, we show that for almost all graphs on n vertices, their boxicity is O(d av ln n) where d av is the average degree.

Efficient Colored Orthogonal Range Counting

Let P be a set of n points in $\mathbb{R}^d$, so that each point is colored by one of C given colors. We present algorithms for preprocessing P into a data structure that efficiently supports queries of the following form: Given an axis-parallel box Q, count the number of distinct colors of the points of $P\cap Q$. We present a general and relatively simple solution that has a polylogarithmic query time and worst-case storage about $O(n^d)$. It is based on several interesting structural properties of the problem, which we establish here. We also show that for random inputs, the data structure requires almost linear expected storage. We then present several techniques for achieving space-time tradeoff. In $\mathbb{R}^2$, the most efficient solution uses fast matrix multiplication in the preprocessing stage. In higher dimensions we use simpler tradeoff mechanisms, which behave just as well. We give a reduction from matrix multiplication to the off-line version of problem, which shows that in $\mathbb{R}^2$ our time-space tradeoffs are reasonably sharp, in the sense that improving them substantially would improve the best exponent of matrix multiplication. Finally, we present a generalized matrix multiplication problem and show its intimate relation to counting colors in boxes in higher dimension.

The cubicity of hypercube graphs

For a graph G , its cubicity cub ( G ) is the minimum dimension k such that G is representable as the intersection graph of (axis-parallel) cubes in k -dimensional space. (A k -dimensional cube is a Cartesian product R 1 × R 2 × ⋯ × R k , where R i is a closed interval of the form [ a i , a i + 1 ] on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113–118] showed that for a d -dimensional hypercube H d , d − 1 log d ≤ cub ( H d ) ≤ 2 d . In this paper, we use the probabilistic method to show that cub ( H d ) = Θ ( d log d ) . The parameter boxicity generalizes cubicity: the boxicity box ( G ) of a graph G is defined as the minimum dimension k such that G is representable as the intersection graph of axis-parallel boxes in k -dimensional space. Since box ( G ) ≤ cub ( G ) for any graph G , our result implies that box ( H d ) = O ( d log d ) . The problem of determining a non-trivial lower bound for box ( H d ) is left open.

Active sampling for multiple output identification

We study functions with multiple output values, and use active sampling to identify an example for each of the possible output values. Our results for this setting include: (1) Efficient active sampling algorithms for simple geometric concepts, such as intervals on a line and axis parallel boxes. (2) A characterization for the case of binary output value in a transductive setting. (3) An analysis of active sampling with uniform distribution in the plane. (4) An efficient algorithm for the Boolean hypercube when each output value is a monomial.

Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy

In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d -dimensional unit cube. In the second part we apply these results to geometric discrepancy. We derive upper bounds for the inverse of the star and the extreme discrepancy with explicitly given small constants and an optimal dependence on the dimension d , and provide corresponding bounds for the star and the extreme discrepancy itself. These bounds improve known results from [B. Doerr, M. Gnewuch, A. Srivastav, Bounds and constructions for the star-discrepancy via δ -covers, J. Complexity 21 (2005) 691–709], [M. Gnewuch, Bounds for the average L p -extreme and the L ∞ -extreme discrepancy, Electron. J. Combin. 12 (2005) Research Paper 54] and [H. N. Mhaskar, On the tractability of multivariate integration and approximation by neural networks, J. Complexity 20 (2004) 561–590]. We also discuss an algorithm from [E. Thiémard, An algorithm to compute bounds for the star discrepancy, J. Complexity 17 (2001) 850–880] to approximate the star-discrepancy of a given n -point set. Our lower bound on the bracketing number of anchored boxes, e.g., leads directly to a lower bound of the running time of Thiémard's algorithm. Furthermore, we show how one can use our results to modify the algorithm to approximate the extreme discrepancy of a given set.

Bounds for the Average $L^p$-Extreme and the $L^\infty$-Extreme Discrepancy

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the $d$-dimensional unit cube with respect to the set system of axis-parallel boxes. For $2\leq p < \infty$ we provide upper bounds for the average $L^p$-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the $L^\infty$-extreme discrepancy with optimal dependence on the dimension $d$ and explicitly given constants.

On the cubicity of certain graphs

The boxicity of a graph G is the minimum dimension b such that G is representable as the intersection graph of axis-parallel boxes in the b-dimensional space. When the boxes are restricted to be axis-parallel b-dimensional cubes, the minimum dimension b required to represent G is called the cubicity of G. In this paper we show that cubicity ( H d ) ⩽ 2 d , where H d is the d-dimensional hypercube. (The d-dimensional hypercube is the graph on 2 d vertices which corresponds to the 2 d d-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate.) We also show that cubicity ( H d ) ⩾ ( d − 1 ) / ( log d ) . We also show that (1) cubicity ( G ) ⩾ ( log α ) / ( log ( D + 1 ) ) , (2) cubicity ( G ) ⩾ ( log n − log ω ) / ( log D ) , where α , ω , D and n denote the stability number, the clique number, the diameter and the number of vertices of G. As consequences of these lower bounds we provide lower bounds for the cubicity of planar graphs, bipartite graphs, triangle-free graphs, etc., in terms of their diameter and the number of vertices.

Planar Maximum Box Problem

Given two finite sets of points X + and X − in ℝ d , the maximum box problem asks to find an axis-parallel box B such that B∩X −=∅ and the total number of points from X + covered is maximized. In this paper we consider the version of the problem for d = 2 (and find the smallest solution box). We present an O(n 3 log4 n) runtime algorithm, thus improving previously best known solution by almost quadratic factor.

Box-Trees and R-Trees with Near-Optimal Query Time

A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. Finally, we present algorithms to convert box-trees to R-trees, resulting in R-trees with (almost) optimal query complexity.

A partition theorem of Tverberg-type for boxes in R3

The following analogue of Tverberg's partition theorem in R 3 is proved (among others): Every set of ⌊ 18 7 m⌋ points in R 3 can be divided into m subsets such that the smallest axis-parallel boxes containing the subsets have a common point.

The Discrepancy of Boxes in Higher Dimension

We prove that the red--blue discrepancy of the set system formed by n points and n axis-parallel boxes in R d can be as high as n Ω(1) in any dimension d= Ω(log n) . This contrasts with the fixed-dimensional case d=O(1) , where the discrepancy is always polylogarithmic. More generally we show that in any dimension 1 R d . We prove that the number of arithmetic operations is at least Ω(nd+ nlog log n) in any dimension d=O(log n) .