Results 1  10
of
13
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
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Cited by 46 (3 self)
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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
Random Sampling, Halfspace Range Reporting, and Construction of (≤k)Levels in Three Dimensions
 SIAM J. COMPUT
, 1999
"... Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the co ..."
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Cited by 33 (7 self)
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Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the orderk Voronoi diagram in O(n log n + nk log k) expected time.
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 ..."
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Cited by 30 (7 self)
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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 ..."
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Cited by 23 (3 self)
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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.
Kinetic Medians and kdTrees
, 2002
"... We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor. ..."
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Cited by 22 (8 self)
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We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor.
Remarks on kLevel Algorithms in the Plane
, 1999
"... In light of recent developments, this paper reexamines the fundamental geometric problem of how to construct the klevel in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decadeold sweepline algorithm by Edelsbrunner and ..."
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Cited by 12 (6 self)
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In light of recent developments, this paper reexamines the fundamental geometric problem of how to construct the klevel in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decadeold sweepline algorithm by Edelsbrunner and Welzl, which now runs in O(n log m+m log 1+" n) deterministic time and O(n) space, where m is the output size and " is any positive constant. We discuss simplification of the data structure in this particular application, by viewing the problem kinetically. ffl HarPeled recently announced a randomized algorithm with an expected running time of O((n + m)ff(n) log n). We observe that a version of an earlier randomized incremental algorithm by Agarwal, de Berg, Matousek, and Schwarzkopf yields almost the same result. ffl The current combinatorial bound by Dey shows that m = O(nk 1=3 ) in the worst case. We give an algorithm that guarantees O(n log n + nk 1=3 ) expected time. 1 Introd...
Decision making based on approximate and smoothed pareto curves
 In Proc. of 16th ISAAC
, 2005
"... We consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them: (i) computing the Pareto curve and (ii) the socalled decision maker’s approach in which both criteria are combined into a single (usually nonlinear) objective function. Pr ..."
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Cited by 11 (2 self)
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We consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them: (i) computing the Pareto curve and (ii) the socalled decision maker’s approach in which both criteria are combined into a single (usually nonlinear) objective function. Previous work by Papadimitriou and Yannakakis showed how to efficiently approximate the Pareto curve for problems like Shortest Path, Spanning Tree, and Perfect Matching. We wish to determine for which classes of combined objective functions the approximate Pareto curve also yields an approximate solution to the decision maker’s problem. We show that an FPTAS for the Pareto curve also gives an FPTAS for the decision maker’s problem if the combined objective function is growth bounded like a quasipolynomial function. If the objective function, however, shows exponential growth then the decision maker’s problem is NPhard to approximate within any polynomial factor. In order to bypass these limitations of approximate decision making, we turn our attention to Pareto curves in the probabilistic framework of smoothed analysis. We show that in a smoothed model, we can efficiently generate the (complete and exact) Pareto curve with a small failure probability if there exists an algorithm for generating the Pareto curve whose worst case running time is pseudopolynomial. This way, we can solve the decision maker’s problem w.r.t. any nondecreasing objective function for randomly perturbed instances of, e. g.,
On the Area Bisectors of a Polygon
, 1997
"... We consider the family of lines that arearea bisectors of a polygon (possibly with holes) in the plane. We say that two bisectors of a polygon P are combinatorially distinct if they induce different partitionings of the vertices of P . We derive an algebraic characterization of area bisectors. We th ..."
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Cited by 7 (5 self)
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We consider the family of lines that arearea bisectors of a polygon (possibly with holes) in the plane. We say that two bisectors of a polygon P are combinatorially distinct if they induce different partitionings of the vertices of P . We derive an algebraic characterization of area bisectors. We then show that there are simple polygons with n vertices that have # n 2 # combinatorially distinct area bisectors #matching the obvious upper bound#, and present an outputsensitive algorithm for computing an explicit representation of all the bisectors of a given polygon. Our study is motivated by the development of novel, flexible feeding devices for parts positioning and orienting. The question of determining all the bisectors of polygonal parts arises in connection with the development of efficient part positioning strategies when using these devices.
Finding an optimal path without growing the tree
 6th Annual European Sym. on Algorithms, Springer LNCS
, 1998
"... For problems on computing an optimal path as well as its length in a certain setting, the “standard” approach for finding an actual optimal path is by building (or “growing”) a singlesource optimal path tree. In this paper, we study a class of optimal path problems with the following phenomenon: Th ..."
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Cited by 3 (0 self)
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For problems on computing an optimal path as well as its length in a certain setting, the “standard” approach for finding an actual optimal path is by building (or “growing”) a singlesource optimal path tree. In this paper, we study a class of optimal path problems with the following phenomenon: The space complexity of the algorithms for reporting the lengths of singlesource optimal paths for these problems is asymptotically smaller than the space complexity of the “standard ” treegrowing algorithms for finding actual optimal paths. We present a general and efficient algorithmic paradigm for finding an actual optimal path for such problems without having to grow a singlesource optimal path tree. Our paradigm is based on the “marriagebeforeconquer ” strategy, the pruneandsearch technique, and a new data structure called clipped trees. The paradigm enables us to compute an actual path for a number of optimal path problems and dynamic programming problems in computational geometry, graph theory, and combinatorial optimization. Our algorithmic