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Shellable nonpure complexes and posets. I
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of ..."
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Cited by 137 (7 self)
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The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed fvectors and hvectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their StanleyReisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the kequal partition lattice (the intersection lattice of the kequal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the kequal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
A Library for Doing Polyhedral Operations
, 1993
"... Polyhedra are geometric representations of linear systems of equations and inequalities. Since polyhedra are used to represent the iteration domains of nested loop programs, procedures for operating on polyhedra are useful for doing loop transformations and other program restructuring transformatio ..."
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Cited by 123 (13 self)
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Polyhedra are geometric representations of linear systems of equations and inequalities. Since polyhedra are used to represent the iteration domains of nested loop programs, procedures for operating on polyhedra are useful for doing loop transformations and other program restructuring transformations which are needed in parallelizing compilers. Thus a need for a library of polyhedral operations has recently been recognized in the parallelizing compiler community. Polyhedra are also used in the definition of domains of variables in systems of affine recurrence equations (SARE). Alpha is a language which is based on the SARE formalism in which all variables are declared over finite unions of polyhedra. This report describes a library of polyhedral functions which was developed to support the Alpha language environment, and which is general enough to satisfy the needs of researchers doing parallelizing compilers. This report describes the data structures used to represent domains, gives...
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 97 (9 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 73 (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...
The octahedron abstract domain
 In Static Analysis Symposium (2004
, 2004
"... NOTICE: This is the author’s version of a work that was accepted for publication in Science of Computer Programming. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this docu ..."
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Cited by 55 (1 self)
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NOTICE: This is the author’s version of a work that was accepted for publication in Science of Computer Programming. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. A definitive version was subsequently published in Science of Computer Programming, 64(2007):115139.
On Bregman Voronoi Diagrams
 in "Proc. 18th ACMSIAM Sympos. Discrete Algorithms
, 2007
"... The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi ..."
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Cited by 48 (23 self)
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The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a byproduct, Bregman Voronoi diagrams allow one to define informationtheoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other
A New Index for Polytopes
"... A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized DehnSommerville equations. It can be computed via a shelling of the polytope. The ranks of t ..."
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Cited by 40 (0 self)
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A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized DehnSommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual.
Efficient Approximation and Optimization Algorithms for Computational Metrology
 PROC. 8TH ACMSIAM SYMPOS. DISCRETE ALGORITHMS
, 1997
"... We give efficient algorithms for solving several geometric problems in computational metrology, focusing on the fundamental issues of "flatness" and "roundness." Specifically, we give approximate and exact algorithms for 2 and 3dimensional roundness primitives, deriving results ..."
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Cited by 40 (3 self)
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We give efficient algorithms for solving several geometric problems in computational metrology, focusing on the fundamental issues of "flatness" and "roundness." Specifically, we give approximate and exact algorithms for 2 and 3dimensional roundness primitives, deriving results that improve previous approaches in several respects, including problem definition, running time, underlying computational model, and dimensionality of the input. We also study methods for determining the width of a ddimensional point set, which corresponds to the metrology notion of "flatness," giving an approximation method that can serve as a fast exactcomputation filter for this metrology primitive. Finally, we report on experimental results derived from implementation and testing, particularly in 3space, of our approximation algorithms, including several heuristics designed to significantly speedup the computations in practice.