6 research outputs found
Bringing Order to Special Cases of Klee's Measure Problem
Klee's Measure Problem (KMP) asks for the volume of the union of n
axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm
has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known
for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP
(where all boxes are cubes of equal side length), Hypervolume (where all boxes
share a vertex), and k-Grounded (where the projection onto the first k
dimensions is a Hypervolume instance).
In this paper we bring some order to these special cases by providing
reductions among them. In addition to the trivial inclusions, we establish
Hypervolume as the easiest of these special cases, and show that the runtimes
of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we
show that any algorithm for one of the special cases with runtime T(n,d)
implies an algorithm for the general case with runtime T(n,2d), yielding the
first non-trivial relation between KMP and its special cases. This allows to
transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)}
algorithm exists for any of the special cases under reasonable complexity
theoretic assumptions. Furthermore, assuming that there is no improved
algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 -
eps})) this reduction shows that there is no algorithm with runtime
O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same
assumption we show a tight lower bound for a recent algorithm for 2-Grounded
[Yildiz,Suri'12].Comment: 17 page
Computing Volumes and Convex Hulls: Variations and Extensions
Geometric techniques are frequently utilized to analyze and reason about multi-dimensional data. When confronted with large quantities of such data, simplifying geometric statistics or summaries are often a necessary first step. In this thesis, we make contributions to two such fundamental concepts of computational geometry: Klee's Measure and Convex Hulls. The former is concerned with computing the total volume occupied by a set of overlapping rectangular boxes in d-dimensional space, while the latter is concerned with identifying extreme vertices in a multi-dimensional set of points. Both problems are frequently used to analyze optimal solutions to multi-objective optimization problems: a variant of Klee's problem called the Hypervolume Indicator gives a quantitative measure for the quality of a discrete Pareto Optimal set, while the Convex Hull represents the subset of solutions that are optimal with respect to at least one linear optimization function.In the first part of the thesis, we investigate several practical and natural variations of Klee's Measure Problem. We develop a specialized algorithm for a specific case of Klee's problem called the āgroundedā case, which also solves the Hypervolume Indicator problem faster than any earlier solution for certain dimensions. Next, we extend Klee's problem to an uncertainty setting where the existence of the input boxes are defined probabilistically, and study computing the expectation of the volume. Additionally, we develop efficient algorithms for a discrete version of the problem, where the volume of a box is redefined to be the cardinality of its overlap with a given point set.The second part of the thesis investigates the convex hull problem on uncertain input. To this extent, we examine two probabilistic uncertainty models for point sets. The first model incorporates uncertainty in the existence of the input points. The second model extends the first one by incorporating locational uncertainty. For both models, we study the problem of computing the probability that a given point is contained in the convex hull of the uncertain points. We also consider the problem of finding the most likely convex hull, i.e., the mode of the convex hull random variable
Computing the volume of the union of cubes
Let C be a set of n axis-aligned cubes in R 3, and let U(C) denote the union of C. We present an algorithm that can compute the volume of U(C) in time O(n 4/3 log n). The previously best known algorithm was by Overmars and Yap, which computes the volume of the union of boxes in R 3 in O(n 3/2 log n) time
ABSTRACT Computing the Volume of the Union of Cubes ā
Let C be a set of n axis-aligned cubes in R 3, and let U(C) denote the union of C. We present an algorithm that can compute the volume of U(C) in time O(n 4/3 log n). The previously best known algorithm, by Overmars and Yap, computes the volume of the union of any n axis-aligned boxes in R 3 in O(n 3/2 log n) time