40171838
Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes
A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes.
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.
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.
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.
46528104
Optimal Tetrahedralizations of Some Convex Polyhedra
325991
Online simultaneous localization and mapping with detection and tracking of moving objects: theory and results from a ground vehicle in crowded urban areas
The simultaneous localization and mapping (SLAM) with detection and tracking of moving objects (DATMO) problem is not only to solve the SLAM problem in dynamic environments but also to detect and track these dynamic objects. In this paper, we derive the Bayesian formula of the SLAM with DATMO problem, which provides a solid basis for understanding and solving this problem. In addition, we provide a practical algorithm for performing DATMO from a moving platform equipped with range sensors. The probabilistic approach to solve the whole problem has been implemented with the Navlab11 vehicle. More than 100 miles of experiments in crowded urban areas indicated that SLAM with DATMO is indeed feasible.
10710066
An Algorithm for the Chromatic Number of a Graph
Christofides' algorithm for finding the chromatic number of a graph is improved both in speed and memory space by using a depth-first search rule to search for a shortest path in a reduced subgraph tree.
9741529
Mining frequent itemsets in distributed and dynamic databases
Traditional methods for frequent itemset mining typically assume that data is centralized and static. Such methods impose excessive communication overhead when data is distributed, and they waste computational resources when data is dynamic. We present what we believe to be the first unified approach that overcomes these assumptions. Our approach makes use of parallel and incremental techniques to generate frequent itemsets in the presence of data updates without examining the entire database, and imposes minimal communication overhead when mining distributed databases. Further, our approach is able to generate both local and global frequent itemsets. This ability permits our approach to identify high-contrast frequent itemsets, which allows one to examine how the data is skewed over different sites.
46393116
Learning Approximate MRFs from Large Transaction Data
In this paper we consider the problem of learning approximate Markov Random Fields (MRFs) from large transaction data. We rely on frequent itemsets to learn MRFs on the data. Since learning exact large MRFs is generally intractable, we resort to learning approximate MRFs. Our proposed modeling approach first employs graph partitioning to cluster variables into balanced disjoint partitions, and then augments important interactions across partitions to capture interdependencies across them. A novel treewidth based augmentation scheme is proposed to boost performance. We learn an exact local MRF for each partition and then combine all the local MRFs together to derive a global model of the data. A greedy approximate inference scheme is developed on this global model. We demonstrate the use of the learned MRFs on the selectivity estimation problem. Empirical evaluation on real datasets demonstrates the advantage of our approach over extant solutions.
8882975
Locating Sensors in Concave Areas
In sensor network localization, multihop based approaches were proposed to approximate the shortest paths to Euclidean distances between pairwise sensors. A good approximation can be achieved when sensors are densely deployed in a convex area, where the shortest paths are close to straight lines connecting pairwise sensors. However, in a concave network, the shortest paths may deviate far away from straight lines, which leads to erroneous distance estimation and inaccurate localization results. In this paper, we propose an improved multihop algorithm which can recognize and filter out the erroneous distance estimation, and therefore achieve accurate localization results even in a concave network.