Weintroduce a new method for finding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex k-vertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area k-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.

We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k-point subsets, matching point sets under translation, computing rectilinear p-centers and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm rand-min 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a well-known fact [27, 44], the expected number of times that step 5 is execut...

We investigate the problem of optimal resource allocation for end-to-end QoS requirements on unicast paths and multicast trees. Specifically, we consider a framework in which resource allocation is based on local QoS requirements at each network link, and associated with each link is a cost function that increases with the severity of the QoS requirement. Accordingly, the problem that we address is how to partition an end-to-end QoS requirement into local requirements, such that the overall cost is minimized. We establish efficient (polynomial) solutions for both unicast and multicast connections. These results provide the required foundations for the corresponding QoS routing schemes, which identify either paths or trees that lead to minimal overall cost. In addition, we show that our framework provides better tools for coping with other fundamental multicast problems, such as dynamic tree maintenance. Keywords --- QoS, QoS-dependent costs, Multicast, Routing, Broadband ne...

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...

We consider the p-piercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or near-linear algorithms for small values of p in cases where the given regions are either axisparallel rectangles or convex c-oriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions) . We also investigate the planar rectilinear (and polygonal) p-center problem, in which we are given a set S of n points in the plane, and wish to find p axis-parallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a linear-time solution for the rectilinear 3-center problem (by showing that this problem can be formulated as an LP-type problem and by exhibiting a relation to Helly numbers). We give O(n log n...

A large variety of problems in computer science can be viewed from a common viewpoint as instances of "algebraic" path problems. Among them are of course path problems in graphs such as the shortest path problem or problems of finding optimal paths with respect to more generally defined objective functions; but also graph problems whose formulations do not directly involve the concept of a path, such as finding all bridges and articulation points of a graph; Moreover, there are even problems which seemingly have nothing to do with graphs, such as the solution of systems of linear equations, partial differentiation, or the determination of the regular expression describing the language accepted by a finite automaton. We describe the relation among these problems and their common algebraic foundation. We survey algorithms for solving them: vertex elimination algorithms such as Gauß-Jordan elimination; and iterative algorithms such as the "classical" Jacobi and Gauß-Seidel iteration.

Our Continuous Adaptive Monitoring (CAM) system provides responses for continuous queries by monitoring and extracting information scattered across the web. Continuous queries are the queries for which responses given to users must be continuously updated, as the sources of interest get updated. Such queries occur, for instance, during on-line decision making, e.g., traffic flow control, weather monitoring etc. Whereas push-based techniques may be able to refresh query results meeting user requirements, they do not scale well. With the pull based approach, the problem of keeping the responses current reduces to the problem of deciding how often to visit a source to determine if and how it has been modified so that a user response can be updated accordingly. As should be evident, periodical monitoring is not scalable. Also, it can lead to huge wastage of monitoring resources. Hence CAM employs a multiphase approach. In the tracking phase, changes to an initially identified set of relevant pages, are tracked. From the observed change characteristics of these pages, a probabilistic model of their change behaviour is formulated and weights are assigned to pages to denote their importance for the current queries. Based on these statistics, during the next phase, the Resource Allocation phase, resources, needed to continuously monitor these pages for changes, are allocated. Given these resource allocations, the Scheduling phase produces an optimal achievable schedule of monitoring. An experimental evaluation of our approach compared to prior approaches on synthetic data for crawling dynamic web pages shows the effectiveness of our approach to monitoring dynamic changes. For example, by monitoring just 5 % of the possible change instances, CAM is able to return 90 % of the changed information to the users. In this demonstration, we show how CAM keeps users up-to-date with respect to a set of ongoing sports related events. 1.

This thesis develops the idea of probabilistic arithmetic. The aim is to replace arithmetic operations on numbers with arithmetic operations on random variables. Specifically, we are interested in numerical methods of calculating convolutions of probability distributions. The long-term goal is to be able to handle random problems (such as the determination of the distribution of the roots of random algebraic equations) using algorithms which have been developed for the deterministic case. To this end, in this thesis we survey a number of previously proposed methods for calculating convolutions and representing probability distributions and examine their defects. We develop some new results for some of these methods (the Laguerre transform and the histogram method), but ultimately find them unsuitable. We find that the details on how the ordinary convolution equations are calculated are

Given an n-point set, the problems of enumerating the k closest pairs and selecting the k-th smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixed-dimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n + n 2=3 k 1=3 log 5=3 n). We also describe output-sensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search. Keywords: distance enumeration, distance selection, closest pairs, range counting, randomized algorithms. 1 Introduction Finding the closest pair of an n-point set has a long history in computational geometry (see [34] for a nice survey). In the plane, the problem can be solved in O(n log n) time using the Delaunay triangulation. In an arbitrary fixed dimension d, the first O(n log n) algorithm, based on di...

In this paper we consider the following covering problem. Given a set S of n points in d-dimensional space, d 2, find two axis-parallel boxes that together cover the set S such that the measure of the largest box is minimized, where the measure is a monotone function of the box. We present a simple algorithm for finding boxes in O(n log n + n d\Gamma1 ) time and O(n) space. Key words: Algorithms, Computational geometry, optimization, axis-parallel 1 Introduction We consider the following min-max two box problem. Given a set S of n points in d-dimensional space, d 2, find two axis-parallel boxes b 1 and b 2 that together cover the set S and minimize the maximum of measures (b 1 ) and (b 2 ), where is a monotone function of the box, i.e. b 1 ` b 2 implies (b 1 ) (b 2 ). Examples of the box measure are the volume of the box, the perimeter of the box (in higher dimensions it can be defined as the sum of 1-dimensional edges or as the area of the boundary of the box), the length of...