We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...

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We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a lineartime deterministic one. The constant of proportionality is d O(d) , which is better than for previously known such algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other problems (such as finding the maximum volume ellipsoid inscribed into the intersection of n halfspaces) in linear time.

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d. A point c ∈ IR d is a β-center point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d + 1)-center point; our algorithm finds an Ω(1/d 2)-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log 2 d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ɛ-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly. 1

Improving on a recent breakthrough of Sharir, we find two minimum-radius circular disks covering a planar point set, in randomized expected time O(n log 2 n). 1 Introduction The k-center problem for a point set S is to find k points (called centers, usually not required to be a subset of S) such that the maximum distance from any point in S to the nearest center is minimized. A case of particular interest is the planar two-center problem [4], which can be viewed less abstractly as one of covering a set of points in the plane by two congruent circular disks, in such a way as to minimize the radius r # of the disks. For a long time the best algorithms for this problem had time bounds of the form O(n 2 log c n) [1, 5, 12, 11]. In a recent breakthrough, Sharir [16] greatly improved all of these algorithms, giving a two-center algorithm with running time O(n log c n). The basic idea is to search for different types of partition depending on the relative positions of the two disk...

. Partitioning a multi-dimensional data set into rectangular partitions subject to certain constraints is an important problem that arises in many database applications, including histogram-based selectivity estimation, load-balancing, and construction of index structures. While provably optimal and efficient algorithms exist for partitioning one-dimensional data, the multi-dimensional problem has received less attention, except for a few special cases. As a result, the heuristic partitioning techniques that are used in practice are not well understood, and come with no guarantees on the quality of the solution. In this paper, we present algorithmic and complexity-theoretic results for the fundamental problem of partitioning a two-dimensional array into rectangular tiles of arbitrary size in a way that minimizes the number of tiles required to satisfy a given constraint. Our main results are approximation algorithms for several partitioning problems that provably approxima...

An Abstract Optimization Problem (AOP) is a triple (H, <, Φ) where H is a finite set, < a total order on 2 H and Φ an oracle that, for given F ⊆ G ⊆ H, either reports that F = min<{F ′ | F ′ ⊆ G} or returns a set F ′ ⊆ G with F ′ < F. To solve the problem means to find the minimum set in H. We present a randomized algorithm that solves any AOP with an expected number of at most e 2 √ n+O ( 4 √ n ln n) oracle calls, n = |H|. In contrast, any deterministic algorithm needs to make 2 n − 1 oracle calls in the worst case. The algorithm is applied to the problem of finding the distance between two n-vertex (or nfacet) convex polyhedra in d-space, and to the computation of the smallest ball containing n points in d-space; for both problems we give the first subexponential bounds in the arithmetic model of computation.

Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1

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We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the Klee-Minty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability.