Klee's Measure Research Articles

Computing the depth distribution of a set of boxes

Motivated by the analysis of range queries in databases, we introduce the computation of the depth distribution of a set B of n d-dimensional boxes (i.e., axis aligned d-dimensional hyperrectangles), which generalizes the computation of the Klee's measure and maximum depth of B. We present an algorithm to compute the depth distribution running in time within O(nd+12log⁡n), using space within O(nlog⁡n), and refine these upper bound for various measures of difficulty of the input instances. Moreover, we introduce conditional lower bounds for this problem which not only provide insights on how fast the depth distribution can be computed, but also clarify the relation between the Depth Distribution problem and other fundamental problems in computer science.

A fast implementation for the 2D/3D box placement problem

The box placement problem involves finding a location to place a rectangular box into a container given n rectangular boxes that have already been placed. It commonly arises as a subproblem in many algorithms for cutting stock problems as well as 2D/3D packing problems. We show that the box placement problem is closely related to some well-studied problems in computational geometry, such as the maximum depth problem and Klee's measure problem. This allows us to leverage on existing techniques for these problems to develop new algorithms for the box placement problem that are not only conceptually simpler but also asymptotically fastest for 2D and faster than existing approaches for 3D. Our implementations rely on augmenting the standard segment tree for 2D or quadtree for 3D, and can be directly incorporated as subroutines into many algorithms for cutting and packing problems.

Efficient transformations for Klee's measure problem in the streaming model

Given a stream of rectangles over a discrete space, we consider the problem of computing the total number of distinct points covered by the rectangles. This is the discrete version of the two-dimensional Klee's measure problem for streaming inputs. Given 0<ϵ,δ<1, we provide (ϵ,δ)-approximations for bounded side length rectangles and for bounded aspect ratio rectangles. For the case of arbitrary rectangles, we provide an O(log⁡U)-approximation, where U is the total number of discrete points in the two-dimensional space. The time to process each rectangle and the total required space are polylogarithmic in U. The time to answer a query for the total area is constant. We construct efficient transformation techniques that project rectangle areas to one-dimensional ranges and then use a streaming algorithm for the one-dimensional Klee's measure problem to obtain these approximations. The projections are deterministic, and to our knowledge, these are the first approaches of this kind that provide efficiency and accuracy trade-offs in the streaming model.

Maximum-weight planar boxes in [formula omitted] time (and better)

Given a set P of n points in Rd, where each point p of P is associated with a weight w(p) (positive or negative), the Maximum-Weight Box problem is to find an axis-aligned box B maximizing ∑p∈B∩Pw(p). We describe algorithms for this problem in two dimensions that run in the worst case in O(n2) time, and much less on more specific classes of instances. In particular, these results imply similar ones for the Maximum Bichromatic Discrepancy Box problem. These improve by a factor of Θ(lgn) on the previously known worst-case complexity for these problems, O(n2lgn) (Cortés et al., 2009 [9]; Dobkin et al., 1996 [10]). Although the O(n2) result can be deduced from new results on Klee's Measure problem (Chan, 2013 [7]), it is a more direct and simplified (non-trivial) solution. We exploit the connection with Klee's Measure problem to further show that (1) the Maximum-Weight Box problem can be solved in O(nd) time for any constant d≥2; (2) if the weights are integers bounded by O(1) in absolute values, or weights are +1 and −∞ (as in the Maximum Bichromatic Discrepancy Box problem), the Maximum-Weight Box problem can be solved in O((nd/lgdn)(lglgn)O(1)) time; (3) it is unlikely that the Maximum-Weight Box problem can be solved in less than nd/2 time (ignoring logarithmic factors) with current knowledge about Klee's Measure problem.

Computing Klee’s Measure of Grounded Boxes

A well-known problem in computational geometry is Klee's measure problem, which asks for the volume of a union of axis-aligned boxes in d-space. In this paper, we consider Klee's measure problem for the special case where a 2-dimensional orthogonal projection of all the boxes has a common corner. We call such a set of boxes 2-grounded and, more generally, a set of boxes is k-grounded if in a k-dimensional orthogonal projection they share a common corner. Our main result is an O(n (d?1)/2log2 n) time algorithm for computing Klee's measure for a set of n 2-grounded boxes. This is an improvement of roughly $O(\sqrt{n})$ compared to the fastest solution of the general problem. The algorithm works for k-grounded boxes, for any k?2, and in the special case of k=d, also called the hypervolume indicator problem, the time bound can be improved further by a logn factor. The key idea of our technique is to reduce the d-dimensional problem to a semi-dynamic weighted volume problem in dimension d?2. The weighted volume problem requires solving a combinatorial problem of maintaining the sum of ordered products, which may be of independent interest.

Approximating the volume of unions and intersections of high-dimensional geometric objects

We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies, we give a fast FPRAS for all objects where one can (1) test whether a given point lies inside the object, (2) sample a point uniformly, and (3) calculate the volume of the object in polynomial time. It suffices to be able to answer all three questions approximately. We show that this holds for a large class of objects. It implies that Klee's measure problem can be approximated efficiently even though it is #P-hard and hence cannot be solved exactly in polynomial time in the number of dimensions unless P = NP . Our algorithm also allows to efficiently approximate the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time 2 d 1 − ε -approximation for certain boxes unless NP = BPP , but give a simple additive polynomial-time ε-approximation.

S-Metric Calculation by Considering Dominated Hypervolume as Klee's Measure Problem

The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a fast determination of the S-metric value is of essential importance. This work describes how to consider the S-metric as a special case of a more general geometric problem called Klee's measure problem (KMP). For KMP, an algorithm exists with runtime O(n log n + n(d/2) log n), for n points of d >or= 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications realize the algorithm without complex data structures and establish an upper bound of O(n(d/2) log n) for the S-metric calculation for d >or= 3. The performance of the new algorithm is studied in comparison to another state of the art algorithm on a set of academic test functions.

On the Complexity of Computing the Hypervolume Indicator

The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klee's measure problem. In general, Klee's measure problem can be solved with O(n logn + nd/2logn) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than Omega(n log n) can be proven. In this paper, we derive a lower bound of Omega(n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d > 1 by reducing the so-called uniformgap problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set.

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.

An in-place algorithm for Klee's measure problem in two dimensions

We present the first in-place algorithm for solving Klee's measure problem for a set of n axis-parallel rectangles in the plane. Our algorithm runs in O ( n 3 / 2 log n ) time and uses O ( 1 ) extra words in addition to the space needed for representing the input. The algorithm is surprisingly simple and thus very likely to yield an implementation that could be of practical interest. As a byproduct, we develop an optimal algorithm for solving Klee's measure problem for a set of n intervals; this algorithm runs in optimal time O ( n log n ) and uses O ( 1 ) extra space.

Rapid Multiplication of Rectangular Matrices

The number of essential multiplications required to multiply matrices of size $N \times N$ and $N \times N^{0.172} $ is bounded by $CN^2 \log ^2 N$.

The measure problem for rectangular ranges in d-space

Klee recently posed the question: find an efficient algorithm for computing the measure of a set of n intervals on the line, and the analog for n hyperrectangles (ranges) in d-space. The one-dimensional case is easily solved in O( n log n) and Bentley has proved an O( n d−1 log n) algorithm for dimension d ≥ 2. We present an algorithm for Klee's measure problem that has a worst-case running time of only O( n d−1 ) for d⩾3. While Bentley's algorithm is based on segment trees and requires only linear storage for any dimension, the new method is based on quad-trees and requires quadratic storage for d > 2.