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.
7101460
On the Difference Method for Drawing Conic Arcs
Abstract The difference method is an efficient way to generate an inscribed polygon of a conic section. We address the implementation issues of the difference method, that is, how to choose the difference parameter and initial points so that the inscribed polygon has a minimal number of line segments and is within a given tolerance of the conic arc. We also discuss how the difference method can be used to draw conic arcs represented by rational quadratic Bezier curves.
1928330
A near-optimal broadcasting algorithm in all-port wormhole-routed hypercubes
For massively parallel computing mechanism, broadcasting is widely used in a variety of applications. When the computation is distributed among the powerful processors, communication overhead always limits the speedup. To reduce the influence of communication latency, a variety of routing methods have been discussed. In this paper, we propose an efficient routing method of broadcasting on all-port worrnhole-routed hybercubes. By exploiting the distance-insensitivity of wormhole switching, this efficient broadcasting algorithm reduce communication latency to l_rzl_logz(n+l )11steps in an n-dimensional hypercube, which is much better than previous results and very close to the theoretical optimum, [rz/log2(n+l )1.Further, by adopting the Hamming code, a well-known algebraic approach in coding theory, this proposed algorithm is contention-free in each broadcasting step.
7862760
Tagged Probe Interval Graphs
A generalization of interval graph is introduced for cosmid contig mapping of DNA. A graph is a tagged probe interval graph if its vertex set can be partitioned into two subsets of probes and nonprobes, and a closed interval can be assigned to each vertex such that two vertices are adjacent if and only if at least one of them is a probe and one end of its corresponding interval is contained in the interval corresponding to the other vertex. We show that tagged probe interval graphs are weakly triangulated graphs, hence are perfect graphs. For a tagged probe interval graph with a given partition, we give a chordal completion that is consistent to any interval completions with respect to the same vertex partition.
37036367
A New Routing Algorithm for Cyclic Shifts on BRGC Hypercubes
The hypercube is one of the most efficient networks for parallel computation. It can efficiently simulate any other network of the same size. In particular, the N-node hypercube can simulate any O(N)-node array, binary tree, or mesh of trees with only a small constant factor slowdown [3]. Furthermore, it is completely symmetric and decomposable into sub-hypercubes. These properties, in addition to its simplicity, make it be an excellent and popular architecture for distributed memory parallel computers. Arrays are natural (data or process) structures for many applications. They can be embedded onto hypercube nodes either in natural order or in Binary-Reflected Gray Code (BRGC) order. It has been noted [2] that many algorithms use the cyclic shift (or rotation) in a natural way as a primitive step. Hence, efficient implementation of cyclic shifts on hypercube network is vita1 to the performance of hypercube computers. Cyclic shift algorithms for the natural order embedding have been studied in [51 and those for the BRGC embedding have been studied in [2,4]. The BRGC embedding has the property that any two adjacent numbers are mapped to neighboring nodes that can communicate directly through a link. Therefore, it uses only one link to route a message from any node to its sink node when the shift distance is one. In this case, it is superior to the natural order embedding. The existing cyclic shift algorithms for the BRGC embedding usually route a message by correcting successive bits that differ in source and sink nodes from the less significant end to the more significant end or in the other order with some exceptions. However, all the algorithms have their drawbacks. For example, the two algorithms presented by Johnsson [2] are neither link-disjoint. Thus, two routing paths will contend for the same link. This not only implies the need for local buffering to route the conflicting messages but also leads to the slowdown of the performance. The best disjoint link algorithm in the literature is the one proposed by Ouyang and Palem [4], which in the worst case uses $fi links to route a message from any node to its cyclic shifted sink node on an n-dimensional BRGC hypercube. There also have been many researches on the problem of general permutation routing [ll, but their algorithms are all asymptotically efficient and are not suitable for the cyclic shift problem. In this paper, we propose a new routing algorithm for cyclic shifts on BRGC hypercubes based on the divide-and-conquer approach instead of the usual one. It uses at most IZ links to route a message from any node to its cyclic shifted sink node on an n-dimensional BRGC hypercube and
6254649
Disjunctive Image Computation for Embedded Software Verification
Finite state models generated from software programs have unique characteristics that are not exploited by existing model checking algorithms. In this paper, we propose a novel disjunctive image computation algorithm and other simplifications based on these characteristics. Our algorithm divides an image computation into a disjunctive set of easier ones that can be performed in isolation. Hypergraph partitioning is used to minimize the number of live variables in each disjunctive component. We use the live variables to simplify transition relations and reachable state subsets. Our experiments on a set of real-world C programs show that the new algorithm achieves orders-of-magnitude performance improvement over the best known conjunctive image computation algorithm