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47
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 456 (0 self)
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The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.
A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra
, 1992
"... We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following prope ..."
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Cited by 184 (28 self)
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We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties: (a) Virtually no additional storage is required beyond the input data; (b) The output list produced is free of duplicates; (c) The algorithm is extremely simple, requires no data structures, and handles all degenerate cases; (d) The running time is output sensitive for nondegenerate inputs; (e) The algorithm is easy to efficiently parallelize. For example, the algorithm finds the v vertices of a polyhedron in R d defined by a nondegenerate system of n inequalities (or dually, the v facets of the convex hull of n points in R d,where each facet contains exactly d given points) in time O(ndv) and O(nd) space. The v vertices in a simple arrangement of n hyperplanes in R d can be found in O(n 2 dv) time and O(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
How good are convex hull algorithms?
, 1996
"... A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facetinducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are esse ..."
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Cited by 82 (8 self)
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A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facetinducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj + jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability todealwell with "degeneracies," or, inability tocontrol the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fatlattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.
OutputSensitive Results on Convex Hulls, Extreme Points, and Related Problems
, 1996
"... . We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) ..."
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Cited by 65 (13 self)
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. We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 11/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 11/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in ddimen...
Optimal OutputSensitive Convex Hull Algorithms in Two and Three Dimensions
, 1996
"... We present simple outputsensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worstcase optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull. ..."
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Cited by 45 (6 self)
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We present simple outputsensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worstcase optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.
PrimalDual Methods for Vertex and Facet Enumeration
 Discrete and Computational Geometry
, 1998
"... Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration) ..."
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Cited by 33 (7 self)
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Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper, we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (facet, respectively) enumeration problem is the facet (vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper, we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primaldual ...
Variable Independence and Aggregation Closure
 IN ACM SYMPOSIUM ON PRINCIPLES OF DATABASE SYSTEMS
, 1996
"... We discuss the issue of adding aggregation to constraint databases. Previous work has shown that, in general, adding aggregates to constraint databases results in languages that are not closed. We show that by imposing a natural restriction, called variable independence (which is a generalization of ..."
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Cited by 31 (10 self)
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We discuss the issue of adding aggregation to constraint databases. Previous work has shown that, in general, adding aggregates to constraint databases results in languages that are not closed. We show that by imposing a natural restriction, called variable independence (which is a generalization of the assumptions underlying the classical relational model of data) on the schema, we can guarantee that a restricted version of the language with aggregation is closed. We illustrate our approach in the context of linear constraint databases.
New Lower Bounds for Convex Hull Problems in Odd Dimensions
 SIAM J. Comput
, 1996
"... We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follow ..."
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Cited by 26 (7 self)
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We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasisimplicial nvertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that ddimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉hard, in the in the sense of Gajentaan and Overmars.
Generating All Vertices of a Polyhedron Is Hard
 DISCRETE COMPUT GEOM (2008 ) 39 : 174–190 175
, 2008
"... We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other ne ..."
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Cited by 21 (6 self)
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We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P = NP. As a corollary, we solve in the negative two wellknown generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains