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46
A Subexponential Bound for Linear Programming
 ALGORITHMICA
, 1996
"... We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d 2 2 d n),e 2 d ln(n / d)+O ( d+ln n)} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorith ..."
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Cited by 185 (15 self)
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We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d 2 2 d n),e 2 d ln(n / d)+O ( d+ln n)} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying n given linear inequalities in d variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson’s linear programming algorithm, this gives an expected bound of O(d 2 n + e O( √ d ln d) The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of n points in dspace, computing the distance of two nvertex (or nfacet) polytopes in dspace, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).
Approximating extent measure of points
 Journal of ACM
"... We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our tec ..."
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Cited by 117 (28 self)
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We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our technique include�approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms. 1
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 114 (10 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Geometric approximation via coresets
 COMBINATORIAL AND COMPUTATIONAL GEOMETRY, MSRI
, 2005
"... The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q usin ..."
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Cited by 83 (9 self)
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The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q using a relatively inefficient algorithm. The solution for Q is then translated to an approximate solution to the original point set P. This paper describes the ways in which this paradigm has been successfully applied to various optimization and extent measure problems.
Hellytype theorems and generalized linear programming
 DISCRETE COMPUT. GEOM
, 1994
"... This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use the ..."
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Cited by 59 (0 self)
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This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...
A discrete subexponential algorithm for parity games
 STACS’03
, 2003
"... We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly ..."
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Cited by 36 (8 self)
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We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly
Linear Programming, the Simplex Algorithm and Simple Polytopes
 Math. Programming
, 1997
"... In the first part of the paper we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot ru ..."
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Cited by 31 (1 self)
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In the first part of the paper we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a ddimensional polyhedron (briefly, a dpolyhedron) if the points in P affinely span R d . A convex ddimensional polytope (briefly, a dpolytope) is a bounded convex dpolyhedron. Alternatively, a convex dpolytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a dpolyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a kface of P . The empty set and P itself are...
Maintaining Approximate Extent Measures of Moving Points
 IN PROC. 12TH ACMSIAM SYMPOS. DISCRETE ALGORITHMS
, 2001
"... We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smal ..."
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Cited by 31 (4 self)
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We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smallest enclosing disk, width, and smallest area or perimeter bounding rectangle of a set of moving points in R² so that the number of events is only a constant. This contrasts with\Omega\Gamma n²) events that data structures for the maintenance of those exact properties have to handle.
Quasiconvex Analysis of Backtracking Algorithms
, 2003
"... We consider a class of multivariate recurrences frequently arising in the worst case analysis of DavisPutnamstyle exponential time backtracking algorithms for NPhard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to ..."
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Cited by 30 (2 self)
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We consider a class of multivariate recurrences frequently arising in the worst case analysis of DavisPutnamstyle exponential time backtracking algorithms for NPhard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to reduce the problem to that of solving univariate linear recurrences; show how to use quasiconvex programming to determine the weight function yielding the smallest upper bound; and prove that the resulting upper bounds are within a polynomial factor of the true asymptotics of the recurrence. We develop and implement a multiplegradient descent algorithm for the resulting quasiconvex programs, using a realnumber arithmetic package for guaranteed accuracy of the computed worst case time bounds.
Practical Methods for Shape Fitting and Kinetic Data Structures using Core Sets
 In Proc. 20th Annu. ACM Sympos. Comput. Geom
, 2004
"... The notion of εkernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrat ..."
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Cited by 30 (8 self)
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The notion of εkernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrated the significance of εkernel by showing that it yields approximation algorithms for a wide range of geometric optimization problems. We present a simpler and more practical algorithm for computing the εkernel of a set P of points in R d. We demonstrate the practicality of our algorithm by showing its empirical performance on various inputs. We then describe an incremental algorithm for fitting various shapes and use the ideas of our algorithm for computing εkernels to analyze the performance of this algorithm. We illustrate the versatility and practicality of this technique by implementing approximation algorithms for minimum enclosing cylinder, minimumvolume bounding box, and minimumwidth annulus. Finally, we show that εkernels can be effectively used to expedite the algorithms for maintaining extents of moving points. 1