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19
Lower Bounds for Computing Geometric Spanners and Approximate Shortest Paths
 In Proc. 8th Canad. Conf. Comput. Geom
, 1996
"... We consider the problems of constructing geometric spanners, possibly containing Steiner points, for sets of points in the ddimensional space IR d , and constructing spanners and approximate shortest paths among a collection of polygonal obstacles in the plane. The complexities of these problems ..."
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Cited by 22 (7 self)
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We consider the problems of constructing geometric spanners, possibly containing Steiner points, for sets of points in the ddimensional space IR d , and constructing spanners and approximate shortest paths among a collection of polygonal obstacles in the plane. The complexities of these problems are shown to be \Omega\Gamma n log n) in the algebraic computation tree model. Since O(n log n)time algorithms are known for solving these problems, our lower bounds are tight up to a constant factor. 1 Introduction Geometric spanners are data structures that approximate the complete graph on a set of points in the ddimensional space IR d , in the sense that the shortest path (based on such a spanner) between any pair of given points is not more than a factor of t longer than the distance between the points in IR d . Let ø be a fixed constant such that 1 ø 1. Throughout this paper, we measure distances between points in the ddimensional space IR d with the L ø metric, where d ...
An efficient algorithm for computing optimal discrete voltage schedules
 SIAM J. Comput
, 2005
"... Abstract. We consider the problem of job scheduling on a variable voltage processor with d discrete voltage/speed levels. We give an algorithm which constructs a minimum energy schedule for n jobs in O(dn log n) time. Previous approaches solve this problem by first computing the optimal continuous s ..."
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Cited by 20 (1 self)
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Abstract. We consider the problem of job scheduling on a variable voltage processor with d discrete voltage/speed levels. We give an algorithm which constructs a minimum energy schedule for n jobs in O(dn log n) time. Previous approaches solve this problem by first computing the optimal continuous solution in O(n 3) time and then adjusting the speed to discrete levels. In our approach, the optimal discrete solution is characterized and computed directly from the inputs. We also show that O(n log n) time is required; hence the algorithm is optimal for fixed d.
Lower bounds in a parallel model without bit operations
 TO APPEAR IN THE SIAM JOURNAL ON COMPUTING
, 1997
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Sorting Helps for Voronoi Diagrams
 Algorithmica
, 1995
"... It is well known that, using standard models of computation,\Omega\Gamma n log n) time is required to build a Voronoi diagram for n point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. But if the sites are sorted before we start, can the Voronoi diagram ..."
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Cited by 11 (0 self)
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It is well known that, using standard models of computation,\Omega\Gamma n log n) time is required to build a Voronoi diagram for n point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. But if the sites are sorted before we start, can the Voronoi diagram be built any faster? We show that for certain interesting, although nonstandard types of Voronoi diagrams, sorting helps. These nonstandard types of Voronoi diagrams use a convex distance function instead of the standard Euclidean distance. A convex distance function exists for any convex shape, but the distance functions based on polygons (especially triangles) lead to particularly efficient Voronoi diagram algorithms. Specifically, a Voronoi diagram using a convex distance function based on a triangle can be built in O(n log log n) time after initially sorting the n sites twice. Convex distance functions based on other polygons require more initial sorting. 1 Introduction We exam...
Lower Bounds for Fundamental Geometric Problems
 IN 5TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA'97
, 1996
"... We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar question ..."
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Cited by 8 (0 self)
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We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal. Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorit...
On the Least Median Square Problem
, 2003
"... We consider the exact and approximate computational complexity of the multivariate LMS linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in IR and a parameter k, the problem is equivalent to computing the ..."
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Cited by 6 (3 self)
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We consider the exact and approximate computational complexity of the multivariate LMS linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in IR and a parameter k, the problem is equivalent to computing the slab bounded by two parallel hyperplanes of minimum separation that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lower bounds on the computational complexity of these problems.
Maintaining Visibility of a Polygon with a Moving Point of View
 In Proc. 8th Canad. Conf. Comput. Geom
, 1996
"... The following problem is studied in this paper: Given a scene with an nvertex simple polygon and a trajectory path in the plane, construct a data structure for reporting the perspective view from a moving point along the trajectory. We present conceptually simple algorithms for the cases of this pr ..."
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Cited by 5 (2 self)
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The following problem is studied in this paper: Given a scene with an nvertex simple polygon and a trajectory path in the plane, construct a data structure for reporting the perspective view from a moving point along the trajectory. We present conceptually simple algorithms for the cases of this problem in which the trajectory path consists of several line segments or of a conic curve that contains the polygon. Our algorithms take O(n log n) time and O(n) space. We also prove that the problem of reporting perspective views from successive points along a trajectory path takes n log n) time in the worst case in the algebraic computation tree model. Our data structure reports the view from any query point on the trajectory in O(k + log n) time for a view of size k. Keywords: Algorithms, visibility, simple polygon, trajectory, topology change, shortest path. 1 Introduction In this paper, we study the following problem: Given a scene with an nvertex simple polygon P and a trajectory ...
A Lower Bound for the Shortest Path Problem
"... We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrixbased repeated squaring al ..."
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Cited by 5 (0 self)
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We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrixbased repeated squaring algorithm for the Shortest Path Problem is optimal in the unbounded fanin PRAM model without bit operations. 1
On Lower Bounds for the Matrix Chain Ordering Problem
, 1993
"... This paper shows that the radix model is not reasonable for solving the Matrix Chain Ordering Problem. In particular, to have an nmatrix instance of this problem with an optimal parenthesization of depth \Theta(n) in the worst case requires the matrix dimensions to be exponential in n. Considering ..."
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Cited by 2 (2 self)
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This paper shows that the radix model is not reasonable for solving the Matrix Chain Ordering Problem. In particular, to have an nmatrix instance of this problem with an optimal parenthesization of depth \Theta(n) in the worst case requires the matrix dimensions to be exponential in n. Considering bit complexity, a worst case lower bound of \Omega\Gamma n 2 ) is given. This worst case lower bound is parameterized and, depending on the optimal product tree depth, it goes from \Omega\Gamma n 2 ) down to \Omega\Gamma n lg n). Also, this paper gives an \Omega\Gamma n lg n) work lower bound for the Matrix Chain Ordering Problem for a class of algorithms on the atomic comparison model with unit cost comparisons. This lower bound, to the authors' knowledge, captures all known algorithms for solving the Matrix Chain Ordering Problem, but does not consider bit operations. Finally, a tradeoff is given between the bit complexity lower bound and the atomic comparison based lower bound. This...