Results 1  10
of
14
Proximity Problems on Moving Points
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair o ..."
Abstract

Cited by 50 (15 self)
 Add to MetaCart
A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably e#cient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree. The method for maintaining the closest pair of points can be extended to the maintenance of closest pair of other distance functions which allows us to maintain the closest pair of a set of moving objects with similar sizes and of a set of points on a smooth manifold.
On levels in arrangements of lines, segments, planes, and triangles
 Geom
, 1998
"... We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and extending, the wellknown kset problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n p k + 1) for the complexity ..."
Abstract

Cited by 42 (21 self)
 Add to MetaCart
We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and extending, the wellknown kset problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n p k + 1) for the complexity of the kth level in an arrangement of n lines. (b) We derive an improved version of Lov'asz Lemma in any dimension, and use it to prove a new bound, O(n 2
Parametric and Kinetic Minimum Spanning Trees
"... We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * r ..."
Abstract

Cited by 30 (7 self)
 Add to MetaCart
We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * represents time and the graph is subject in addition to changes such as edge insertions, deletions, and modifications of the weight functions as time progresses. We solve both problems in time O(n2=3 log4=3 n) per combinatorial change in the tree (or randomized O(n2=3 log n) per change). Our time bounds reduce to O(n1=2 log3=2 n) per change (O(n1=2 log n) randomized) for planar graphs or other minorclosed families of graphs, and O(n1=4 log3=2 n) per change (O(n1=4 log n) randomized) for planar graphs with weight changes but no insertions or deletions.
On Levels in Arrangements of Curves
 Proc. 41st IEEE
, 2002
"... Analyzing the worstcase complexity of the klevel in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
Analyzing the worstcase complexity of the klevel in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudoparabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudoparabolas into pseudosegments, as well as a new observation for cutting pseudosegments into pieces that can be extended to pseudolines. We mention applications to parametric and kinetic minimum spanning trees.
Improved Bounds on Planar ksets and klevels
 Discrete Comput. Geom
, 1997
"... We prove an O(nk 1=3 ) upper bound for planar ksets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in arrangement ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
We prove an O(nk 1=3 ) upper bound for planar ksets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general. 1 Introduction The problem of determining the optimum asymptotic bound on the number of ksets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer k n, a kset is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains elusive even in ! ...
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
 In Proc. 44th IEEE Sympos. Found. Comput. Sci
, 2003
"... We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degrees polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved klevel results for most of the curve families studied earlier, including a nearO(n ) bound for parabolas.
Robust discrete optimization under ellipsoidal uncertainty sets
, 2004
"... We address the complexity and practically e cient methods for robust discrete optimization under ellipsoidal uncertainty sets. Speci cally, weshowthat the robust counterpart of a discrete optimization problem with correlated objective function data is NPhard even though the nominal problem is polyn ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We address the complexity and practically e cient methods for robust discrete optimization under ellipsoidal uncertainty sets. Speci cally, weshowthat the robust counterpart of a discrete optimization problem with correlated objective function data is NPhard even though the nominal problem is polynomially solvable. For uncorrelated and identically distributed data, however, we show that the robust problem retains the complexity of the nominal problem. For uncorrelated, but not identically distributed data we propose an approximation method that solves the robust problem within arbitrary accuracy. Wealso propose a FrankWolfe type algorithm for this case, whichwe prove converges to a locally optimal solution, and in computational experiments is remarkably e ective. Finally,we propose a generalization of the robust discrete optimization framework we proposed earlier that (a) allows the key parameter that controls the tradeo between robustness and optimality to depend on the solution and (b) results in increased exibility and decreased conservatism, while maintaining the complexity of the nominal problem.
Using Sparsification for Parametric Minimum Spanning Tree Problems
 Nordic J. Computing
, 1996
"... Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning t ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an nnode, medge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...
Lovasz's lemma for the threedimensional klevel of concave surfaces and its applications
 In Proc. 40th IEEE Sympos. Found. Comput. Sci
, 1999
"... 1 Introduction Investigation of the combinatorial properties of klevels of arrangements of curves and surfaces is a central topic in combinatorial geometry. The klevel problem can be considered as the parametric version of the selection problem, which finds the k smallest elements among n weighted ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
1 Introduction Investigation of the combinatorial properties of klevels of arrangements of curves and surfaces is a central topic in combinatorial geometry. The klevel problem can be considered as the parametric version of the selection problem, which finds the k smallest elements among n weighted elements. If we replace the weight of each element by a function with d \Gamma 1 parameters, we have an arrangement of n hypersurfaces in ddimensional space, and its klevel shows the trajectory of the kth smallest element. Note that, in this paper, we count the levels in an arrangement from the bottom, unless explicitly specified. So far, only klevels of arrangements of hyperplanes are investigated well in the literature. Lov'asz [12] gave an O(k1=2n) upper bound for the complexity of the klevel of an arrangement of n lines in the plane by using the following lemma3: