41937900
Detecting tetrahedralizations of a set of line segments
Let L be a set of line segments in three dimensional Euclidean space. In this paper, we prove several characterizations of tetrahedralizations. We present an O(mn log n) algorithm to determine whether L is the edge set of a tetrahedralization, where m is the number of segments and n is the number of distinct endpoints of segments in L. We show that it is NP-complete to decide whether L contains the edge set of a tetrahedralization. We also show that it is NP-complete to decide whether L is a subset of the edge set of a tetrahedralization.
18478446
On Parallel Complexity of Planar Triangulations
The greedy triangulation of a finite planar point set is obtained by repeatedly inserting a shortest diagonal that doesn't intersect those already in the plane. We show that the problem of constructing the greedy triangulation of a finite set of points with integer coordinates in the plane is P-complete. This is the first known geometric P-complete problem where the input is given as a set of points. On the other hand, we provide general NC-methods for testing whether a given triangulation of a set of points and/or line segments can be built by inserting the diagonals in a given partial order, and for constructing such triangulations for simple polygons. As corollaries, we obtain NC-algorithms for testing whether a triangulation is respectively the greedy triangulation or the so called sweep-line triangulation, and for constructing respectively the greedy triangulation or the sweep-line triangulation of a simple polygon. The latter result solves the open problem posed by Atallah et al.
35234330
A Parallel Algorithm for Finding the Constrained Voronoi Diagram of Line Segments in the Plane
In this paper, we present an O(1/αlog n) (for any constant 0 ≤ α ≤ 1) time parallel algorithm for constructing the constrained Voronoi diagram of a set L of n non-crossing line segments in E2, using O(n1+α) processors on a CREW PRAM model. This parallel algorithm also constructs the constrained Delaunay triangulation of L in the same time and processor bound by the duality. 
 
Our method established the conversions from finding the constrained Voronoi diagram L to finding the Voronoi diagram of S, the endpoint set of L. We further showed that this conversion can be done in O(log n) time using n processors in CREW PRAM model. The complexity of the conversion implies that any improvement of the complexity for finding the Voronoi diagram of a point set will automatically bring the improvement of the one in question.
1893362
Minimal Tetrahedralizations of a Class of Polyhedra
Given a simple polyhedron P in the three dimensional Euclidean space, different tetrahedralizations of P may contain different numbers of tetrahedra. The minimal tetrahedralization is a tetrahedralization with the minimum number of tetrahedra. In this paper, we present some properties of the graph of polyhedra. Then we identify a class of polyhedra and show that this kind of polyhedra can be minimally tetrahedralized in O(n2) time.
38065257
Tetrahedralization of Two Nested Convex Polyhedra
In this paper, we present an algorithm to tetrahedralize the region between two nested convex polyhedra without introducing Steiner points. The resulting tetrahedralization consists of linear number of tetrahedra. Thus, we answer the open problem raised by Chazelle and Shouraboura: "whether or not one can tetrahedralize the region between two nested convex polyhedra into linear number of tetrahedra, avoiding Steiner points?". Our algorithm runs in O((n+m)3 log(n+m)) time and produces 2(m+n-3) tetrahedra, where n and m are the numbers of the vertices in the two given polyhedra, respectively.
46528104
Optimal Tetrahedralizations of Some Convex Polyhedra
17451344
Compositional SCC Analysis for Language Emptiness
We propose a refinement approach to language emptiness, which is based on the enumeration and the successive refinements of SCCs on over-approximations of the exact system. Our algorithm is compositional: It performs as much computation as possible on the abstract systems, and prunes uninteresting part of the search space as early as possible. It decomposes the state space disjunctively so that each state subset can be checked in isolation to decide language emptiness for the given system. We prove that the strength of an SCC or a set of SCCs decreases monotonically with composition. This allows us to deploy the proper model checking algorithms according to the strength of the SCC at hand. We also propose to use the approximate distance of a fair cycle from the initial states to guide the search. Experimental studies on a set of LTL model checking problems prove the effectiveness of our method.
1081081
Modelling the learning transaction
A model of learning transactions is offered for use in the development of e-learning materials and toolkits. The model identifies the key components required for effective learning, and in doing so, helps the development team ensure the provision of these components to the teacher or learner.
4374799
Appearance manifolds for modeling time-variant appearance of materials
We present a visual simulation technique called appearance manifolds for modeling the time-variant surface appearance of a material from data captured at a single instant in time. In modeling time-variant appearance, our method takes advantage of the key observation that concurrent variations in appearance over a surface represent different degrees of weathering. By reorganizing these various appearances in a manner that reveals their relative order with respect to weathering degree, our method infers spatial and temporal appearance properties of the material's weathering process that can be used to convincingly generate its weathered appearance at different points in time. Results with natural non-linear reflectance variations are demonstrated in applications such as visual simulation of weathering on 3D models, increasing and decreasing the weathering of real objects, and material transfer with weathering effects.
2143066
Fine-grain abstraction and sequential do not cares for large scale model checking
Abstraction refinement is a key technique for applying model checking to the verification of real-world digital systems. In previous work, the abstraction granularity is often limited at the state variable level, which is too coarse for verifying industrial-scale designs. In this paper, we propose a finer grain abstraction in which intermediate variables are selectively inserted to partition large combinational logic cones into smaller pieces; these intermediate variables, together with the state variables, are then treated as "atoms" in abstraction refinement. With this fine-grain approach, refinement is conducted in two different directions, sequential and Boolean. We propose a SAT-based method for predicting the appropriate refinement direction, and apply greedy minimization in both directions to keep the refinement set small. We also explore the use of approximate reachable states of the remaining submodules to help verifying the abstract model. Experimental studies show that the proposed techniques significantly improve the performance of abstraction refinement, and therefore increase the model checker's ability to handle large designs.
7877505
Three forbidden subgraphs for line graphs
We construct a set of three graphs with the property that a 3-connected graph with minimum degree at least 7 is a line graph if and only if it does not contain any of the three graphs as an induced subgraph. We also show that the number of excluded subgraphs cannot be reduced no matter what connectivity is specified.