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166
An optimal algorithm for intersecting line segments in the plane
 J. ACM
, 1992
"... Abstract. Themain contribution ofthiswork is an O(nlogr ~ +k)timeal gorithmfo rcomputingall k intersections among n line segments in the plane, This time complexity IS easdy shown to be optimal. Within thesame asymptotic cost, ouralgorithm canalso construct thesubdiwslon of theplancdefmed by the se ..."
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Cited by 161 (2 self)
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Abstract. Themain contribution ofthiswork is an O(nlogr ~ +k)timeal gorithmfo rcomputingall k intersections among n line segments in the plane, This time complexity IS easdy shown to be optimal. Within thesame asymptotic cost, ouralgorithm canalso construct thesubdiwslon of theplancdefmed by the segments and compute which segment (if any) lies right above (or below) each intersection and each endpoint. The algorithm has been implemented and performs very well. The storage requirement is on the order of n + k in the worst case, but it is considerably lower in practice. To analyze the complexity of the algorithm, an amortization argument based on a new combinatorial theorem on line arrangements is used.
Bounding the VapnikChervonenkis dimension of concept classes parameterized by real numbers
 Machine Learning
, 1995
"... Abstract. The VapnikChervonenkis (VC) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the VC dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are au ..."
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Cited by 91 (1 self)
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Abstract. The VapnikChervonenkis (VC) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the VC dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are automatic for discrete concept classes, but hitherto little has been known about what general conditions guarantee polynomial bounds on VC dimension for classes in which concepts and examples are represented by tuples of real numbers. In this paper, we show that for two general kinds of concept class the VC dimension is polynomially bounded in the number of real numbers used to define a problem instance. One is classes where the criterion for membership of an instance in a concept can be expressed as a formula (in the firstorder theory of the reals) with fixed quantification depth and exponentiallybounded length, whose atomic predicates are polynomial inequalities of exponentiallybounded degree. The other is classes where containment of an instance in a concept is testable in polynomial time, assuming we may compute standard arithmetic operations on reals exactly in constant time. Our results show that in the continuous case, as in the discrete, the real barrier to efficient learning in the Occam sense is complexitytheoretic and not informationtheoretic. We present examples to show how these results apply to concept classes defined by geometrical figures and neural nets, and derive polynomial bounds on the VC dimension for these classes. Keywords: Concept learning, information theory, VapnikChervonenkis dimension, Milnor’s theorem 1.
New Results on Server Problems
 SIAM Journal on Discrete Mathematics
, 1990
"... In the kserver problem, we must choose how k mobile servers will serve each of a sequence of requests, making our decisions in an online manner. We exhibit an optimal deterministic online strategy when the requests fall on the real line. For the weightedcache problem, in which the cost of moving t ..."
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Cited by 73 (7 self)
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In the kserver problem, we must choose how k mobile servers will serve each of a sequence of requests, making our decisions in an online manner. We exhibit an optimal deterministic online strategy when the requests fall on the real line. For the weightedcache problem, in which the cost of moving to x from any other point is w(x), the weight of x, we also provide an optimal deterministic algorithm. We prove the nonexistence of competitive algorithms for the asymmetric twoserver problem, and of memoryless algorithms for the weightedcache problem. We give a fast algorithm for offline computing of an optimal schedule, and show that finding an optimal offline schedule is at least as hard as the assignment problem. 1 Introduction The kserver problem can be stated as follows. We are given a metric space M , and k servers which move among the points of M , each occupying one point of M . Repeatedly, a request (a point x 2 M) appears. To serve x, each server moves some distance, possibly...
ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 65 (14 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 57 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
MinimumLink Paths Among Obstacles in the Plane
 ALGORITHMICA
, 1992
"... Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a correspon ..."
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Cited by 53 (6 self)
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Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimumlink path) between two points in time O(E#(n) log² n) (and space O(E)), where n is the total number of edges of the obstacles, E is the size of the visibility graph, and #(n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted at s) of minimumlink paths from s to all obstacle vertices. This leads to a method of solving the query version of our problem (for query points t).
Geometric Applications of a Randomized Optimization Technique
 Discrete Comput. Geom
, 1999
"... 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, ..."
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Cited by 53 (6 self)
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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 kpoint subsets, matching point sets under translation, computing rectilinear pcenters 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 randmin 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 wellknown fact [27, 44], the expected number of times that step 5 is execut...
Dynamic planar convex hull
 Proc. 43rd IEEE Sympos. Found. Comput. Sci
, 2002
"... In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage o ..."
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Cited by 53 (1 self)
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In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.
On Bounding the Betti Numbers and Computing the Euler Characteristic of Semialgebraic Sets
, 1996
"... In this paper we give a new bound on the sum of the Betti numbers of semialgebraic sets. This extends a wellknown bound due to Oleinik and Petrovsky [19], Thom [23] and Milnor [18]. In separate papers they proved that the sum of the Betti numbers of a semialgebraic set S ae R k ; defined by P ..."
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Cited by 51 (18 self)
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In this paper we give a new bound on the sum of the Betti numbers of semialgebraic sets. This extends a wellknown bound due to Oleinik and Petrovsky [19], Thom [23] and Milnor [18]. In separate papers they proved that the sum of the Betti numbers of a semialgebraic set S ae R k ; defined by P 1 0; : : : ; P s 0; deg(P i ) d; 1 i s; is bounded by (O(sd)) k : Given a semialgebraic set S ae R k defined as the intersection of a real variety, Q = 0; deg(Q) d; whose real dimension is k 0 ; with a set defined by a quantifierfree Boolean formula with atoms of the form, P i = 0; P i ? 0; P i ! 0; deg(P i ) d; 1 i s; we prove that the sum of the Betti numbers of S is bounded by s k 0 (O(d)) k : In the special case, when S is defined by Q = 0; P 1 ? 0; : : : ; P s ? 0; we have a slightly tighter bound of \Gamma s k 0 \Delta (O(d)) k : This result generalises the OleinikPetrovskyThomMilnor bound in two directions. Firstly, our bound applies to arbitrary semialg...
Linear decision trees: volume estimates and topological bounds
 PROC. 24TH ACM SYMP. ON THEORY OF COMPUTING
, 1992
"... We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a line ..."
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Cited by 49 (5 self)
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We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a linear decision tree for this problem is Θ(nlog(n/k)). The upper bound is easy; the lower bound can be established for k = O(n 1/4−ε) by a volume argument; for the whole range, however, our proof is more complicated and it involves the use of some topology as well as the theory of Möbius functions.