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207
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 81 (17 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm
 POLYTOPES – COMBINATORICS AND COMPUTATION
, 2000
"... This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for ddimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all int ..."
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Cited by 72 (4 self)
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This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for ddimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all integer pivoting. It can also be used to compute the volume of the convex hull of a set of points. For a polyhedron with m inequalities in d variables and known extreme point, it finds all bases in time O(md2) per basis. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex and the complexity reduces to O(md) per vertex). Computational experience is included in the paper, including a comparison with an earlier implementation.
Scalable Custom Instructions Identification for InstructionSet Extensible Processors
 In CASES
, 2004
"... Extensible processors allow addition of applicationspecific custom instructions to the core instruction set architecture. However, it is computationally expensive to automatically select the optimal set of custom instructions. Therefore, heuristic techniques are often employed to quickly search the ..."
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Cited by 66 (8 self)
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Extensible processors allow addition of applicationspecific custom instructions to the core instruction set architecture. However, it is computationally expensive to automatically select the optimal set of custom instructions. Therefore, heuristic techniques are often employed to quickly search the design space. In this paper, we present an efficient algorithm for exact enumeration of all possible candidate instructions given the dataflow graph (DFG) corresponding to a code fragment. Even though this is similar to the “subgraph enumeration” problem (which is exponential), we find that most subgraphs are not feasible candidates for various reasons. In fact, the number of candidates is quite small compared to the size of the DFG. Compared to previous approaches, our technique achieves orders of magnitude speedup in enumerating these candidate custom instructions for very large DFGs.
New Algorithms for Enumerating All Maximal Cliques
, 2004
"... Abstract. In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G = (V, E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M(n)) time delay and in O(n 2) space and the other runs ..."
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Cited by 60 (1 self)
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Abstract. In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G = (V, E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M(n)) time delay and in O(n 2) space and the other runs with O( ∆ 4) time delay and in O(n + m) space, where ∆ denotes the maximum degree of G, M(n) denotes the time needed to multiply two n × n matrices, and the latter one requires O(nm) time as a preprocessing. For a given bipartite graph G, we propose three algorithms for enumerating all maximal bipartite cliques. The first algorithm runs with O(M(n)) time delay and in O(n 2) space, which immediately follows from the algorithm for the nonbipartite case. The second one runs with O( ∆ 3) time delay and in O(n + m) space, and the last one runs with O( ∆ 2) time delay and in O(n + m + N∆) space, where N denotes the number of all maximal bipartite cliques in G and both algorithms require O(nm) time as a preprocessing. Our algorithms improve upon all the existing algorithms, when G is either dense or sparse. Furthermore, computational experiments show that our algorithms for sparse graphs have significantly good performance for graphs which are generated randomly and appear in realworld problems. 1
Expansive motions and the polytope of pointed pseudotriangulations
 Discrete and Computational Geometry  The GoodmanPollack Festschrift, Algorithms and Combinatorics
, 2003
"... We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriang ..."
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Cited by 52 (14 self)
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We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriangulations of the point set and whose edges are flips of interior pseudotriangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an ngon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a byproduct a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of
Discovering Frequent Substructures In Large Unordered Trees
 IN PROC. OF THE 6TH INTL. CONF. ON DISCOVERY SCIENCE
, 2003
"... In this paper, we study a data mining problem of discovering frequent substructures in a large collection of semistructured data, where both of the patterns and the data are modeled by labeled unordered trees. An unordered tree is a directed acyclic graph with a specified node called the root, ..."
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Cited by 51 (6 self)
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In this paper, we study a data mining problem of discovering frequent substructures in a large collection of semistructured data, where both of the patterns and the data are modeled by labeled unordered trees. An unordered tree is a directed acyclic graph with a specified node called the root, and all nodes but the root have at most one parent. Each node is labeled by a symbol drawn from an alphabet. Such unordered trees can be seen as either a generalization of itemsets in relational databases or an efficient specialization of attributed graphs in graph mining. They are also useful in various applications such as analysis of chemical compounds and mining hyperlink structures in Web. Introducing novel definitions of the support and the canonical form for unordered trees, we present an efficient algorithm called Unot that computes all labeled unordered trees appearing in a collection of data trees with frequency above a userspecified threshold. We prove that the algorithm enumerates each frequent pattern T in O(kb n) per pattern, where k is the size of T , b is the branching factor of the data tree, and n is the total number of occurrences of T in the data trees. The keys of the algorithm are e#cient enumerating all unordered trees in canonical form and incrementally computation of the occurrences based on a powerful design technique known as the reverse search
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 38 (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
From the zonotope construction to the Minkowski addition of convex polytopes
 JOURNAL OF SYMBOLIC COMPUTATION
, 2004
"... A zonotope is the Minkowski addition of line segments in R d. The zonotope construction problem is to list all extreme points of a zonotope given by its line segments. By duality, it is equivalent to the arrangement construction problem that is to generate all regions of an arrangement of hyperplane ..."
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Cited by 38 (4 self)
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A zonotope is the Minkowski addition of line segments in R d. The zonotope construction problem is to list all extreme points of a zonotope given by its line segments. By duality, it is equivalent to the arrangement construction problem that is to generate all regions of an arrangement of hyperplanes. By replacing line segments with convex Vpolytopes, we obtain a natural generalization of the zonotope construction problem: the construction of the Minkowski addition of k polytopes. Gritzmann and Sturmfels studied this general problem in various aspects and presented polynomial algorithms for the problem when one of the parameters k or d is fixed. The main objective of the present work is to introduce an efficient algorithm for variable d and k. Here we call an algorithm efficient or polynomial if it runs in time bounded by a polynomial function of both the input size and the output size. The algorithm is a natural extension of a known algorithm for the zonotope construction, based on linear programming and reverse search. It is compact, highly parallelizable and very easy to implement. This work has been motivated by the use of polyhedral computation for optimal tolerance determination in mechanical engineering.
Exact FromRegion Visibility Culling
, 2002
"... To preprocess a scene for the purpose of visibility culling during walkthroughs it is necessary to solve visibility from all the elements of a finite partition of viewpoint space. Many conservative and approximate solutions have been developed that solve for visibility rapidly. The idealised exac ..."
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Cited by 36 (1 self)
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To preprocess a scene for the purpose of visibility culling during walkthroughs it is necessary to solve visibility from all the elements of a finite partition of viewpoint space. Many conservative and approximate solutions have been developed that solve for visibility rapidly. The idealised exact solution for general 3D scenes has often been regarded as computationally intractable. Our exact algorithm for finding the visible polygons in a scene from a region is a computationally tractable preprocess that can handle scenes of the order of millions of polygons. The essence