Line-segment intersection reporting in parallel

In this paper we give a parallel algorithm for line-segment intersection reporting in the plane. It runs in timeO(((n +k) logn log logn)/p) usingp processors on a concurrent-read-exclusive-write (CREW)-PRAM, wheren is the number of line segments,k is the number of intersections, andp ≤n +k.     

Efficient and Scalable PRAM Algorithms for Discrete-Event Simulation of Bounded Degree Networks

We describe two new parallel algorithms, one conservative and another optimistic, for discrete-event simulation on an exclusive-read exclusive-write parallel random-access machine (EREW PRAM). The target physical systems are bounded degree networks which are represented by logic circuits. Employing p processors, our conservative algorithm can simulate up to O(p) independent messages of a system with n logical processes in O(log n) time. The number of processors, p, can be optimally varied in the range 1 ≤ p ≤ n. To identify independent messages, this algorithm also introduces a novel scheme based on a variable size time window. Our optimistic algorithm is designed to reduce the rollback frequency and the memory requirement to save past states and messages. The optimistic algorithm also simulates O(p) earliest messages on a p-processor computer in O(log n) time. To our knowledge, such a theoretical efficiency in parallel simulation algorithms, conservative or optimistic, has been achieved for the first time.     

无向图的边极大匹配并行算法及其应用*

在EREW PRAM(exclusive-read and exclusive-write parallel random access machine)并行计算模型上,对范围很广的一类无向图的边极大匹配问题,给出时间复杂性为O(logn),使用O((n+m)/logn)处理器的最佳、高速并行算法.     

Parallel Dictionaries Using AVL Trees

AVL (Adel'son-Vel'skii and Landis) trees are efficient data structures for implementing dictionaries. We present a parallel dictionary, using AVL trees, on the EREW PRAM by proposing optimal algorithms to performkoperations withp(1 ≤p≤k) processors. An explicit processor scheduling is devised to avoid simultaneous reads in our parallel algorithm to performksearches, which avoids the need for any additional memory in the parallelization. To perform multiple insertions and deletions, we identify rotations (in addition to AVL tree rotations) required to restore balance and present parallel algorithms to performpinsertions/deletions inO(logn+ logp) time withpprocessors.     

An Optimal Shortest Path Parallel Algorithm for Permutation Graphs

We present an optimal parallel algorithm for the single-source shortest path problem for permutation graphs. The algorithm runs in O(log n) time using O(n/log n) processors on an EREW PRAM. As an application, we show that a minimum connected dominating set in a permutation graph can be found in O(log n) time using O(n/log n) processors.     

Reconstructing a Binary Tree from Its Traversals in Doubly Logarithmic CREW Time

We consider the following problem. For a binary tree T = (V, E) where V = {1, 2, ..., n}, given its inorder traversal and either its preorder or its postorder traversal, reconstruct the binary tree. We present a new parallel algorithm for this problem. Our algorithm requires O(n) space. The main idea of our algorithm is to reduce the reconstruction process to merging two sorted sequences. With the best parallel merging algorithms, our algorithm can be implemented in O(log log n) time using O(n/log log n) processors on the CREW PRAM (or in O(log n) time using O(n/log n) processors on the EREW PRAM). Our result provides one more example of a fundamental problem which can be solved by optimal parallel algorithms in O(log log n)time on the CREW PRAM.     

On the geodesic voronoi diagram of point sites in a simple polygon

Given a simple polygon withn sides in the plane and a set ofk point “sites” in its interior or on the boundary, compute the Voronoi diagram of the set of sites using the internal “geodesic” distance inside the polygon as the metric. We describe anO((n + k) log(n + k) logn)-time algorithm for solving this problem and sketch a fasterO((n + k) log(n + k)) algorithm for the case when the set of sites includes all reflex vertices of the polygon in question.     

Efficient parallel algorithms forr-dominating set andp-center problems on trees

We develop efficient parallel algorithms for ther-dominating set and thep-center problems on trees. On a concurrent-read exclusive-write PRAM, our algorithm for ther-dominating set problem runs inO(logn log logn) time withn processors. The algorithm for thep-center problem runs inO(log² n log logn) time withn processors.     

Output-sensitive generation of the perspective view of isothetic parallelepipeds

We present a new hidden-line elemination technique for displaying the perspective view of a scene of three-dimensional isothetic parallelepipeds (3D-rectangles). We assume that the 3D-rectangles are totally ordered based upon the dominance relation of occlusion. The perspective view is generated incrementally, starting with the closest 3D-rectangle and proceeding away from the view point. Our algorithm is scene-sensitive and uses0((n +d) logn log logn) time, wheren is the number of 3D-rectangles andd is the number of edges of the display. This improves over the heretofore best known technique. The primary data structure is an efficient alternative to dynamic fractional cascading for use with augmented segment and range trees when the universe is fixed beforehand. It supports queries inO((logn +k) log logn) time, wherek is the size of the response, and insertions and deletions inO(logn log logn) time, all in the worst case.     

Line-segment intersection reporting in parallel

In this paper we give a parallel algorithm for line-segment intersection reporting in the plane. It runs in timeO(((n +k) logn log logn)/p) usingp processors on a concurrent-read-exclusive-write (CREW)-PRAM, wheren is the number of line segments,k is the number of intersections, andp n +k.This work was supported by the DFG, SFB 124, TP B2, VLSI Entwurfsmethoden und Parallelität.     

Area and Perimeter Computation of the Union of a Set of Iso-rectangles in Parallel

Finding the area and perimeter of the union/intersection of a set of iso-rectangles is a very important part of circuit extraction in VLSI design. We combine two techniques, the uniform grid and the vertex neighborhoods, to develop a new parallel algorithm for the area and perimeter problems which has an average linear time performance but is not worst-case optimal. The uniform grid technique has been used to generate the candidate vertices of the union or intersection of the rectangles. An efficient point-in-rectangles inclusion test filters the candidate set to obtain the relevant vertices of the union or intersection. Finally, the vertex neighborhood technique is used to compute the mass properties from these vertices. This algorithm has an average time complexity of O(((n + k)/p) + log p) where n is the number of input rectangle edges with k intersections on p processors assuming a PRAM model of computation. The analysis of the algorithm on a SIMD architecture is also presented. This algorithm requires very simple data structures which makes the implementation easy. We have implemented the algorithm on a Sun 4/280 workstation and a Connection Machine. The sequential implementation performs better than the optimal algorithm for large datasets. The parallel implementation on a Connection Machine CM-2 with 32K processors also shows good results.     

A parallel algorithm for surface-based object reconstruction

This paper presents a parallel algorithm that approximates the surface of an object from a collection of its planar contours. Such a reconstruction has wide applications in such diverse fields as biological research, medical diagnosis and therapy, architecture, automobile and ship design, and solid modeling. The surface reconstruction problem is transformed into the problem of finding a minimum-cost acceptable path on a toroidal grid graph, where each horizontal and each vertical edge have the same orientation. An acceptable path is closed path that makes a complete horizontal and vertical circuit. We exploit the structure of this graph to develop efficient parallel algorithms for a message-passing computer. Givenp processors and anm byn toroidal graph, our algorithm will find the minimum cost acceptable path inO(mn log(m)/p) steps, ifp =O(mn/((m + n) log(mn/(m + n)))), which is an optimal speedup. We also show that the algorithm will sendO(p ²(m + n)) messages. The algorithm has a linear topology, so it is easy to embed the algorithm in common multiprocessor architectures.     

Parallel Algorithms for the Edge-Coloring and Edge-Coloring Update Problems

LetG(V,E) be a simple undirected graph with a maximum vertex degree Δ(G) (or Δ for short), |V| =nand |E| =m. An edge-coloring ofGis an assignment to each edge inGa color such that all edges sharing a common vertex have different colors. The minimum number of colors needed is denoted by χ′(G) (called thechromatic index). For a simple graphG, it is known that Δ ≤ χ′(G) ≤ Δ + 1. This paper studies two edge-coloring problems. The first problem is to perform edge-coloring for an existing edge-colored graphGwith Δ + 1 colors stemming from the addition of a new vertex intoG. The proposed parallel algorithm for this problem runs inO(Δ^3/2log³Δ + Δ logn) time usingO(max{nΔ, Δ³}) processors. The second problem is to color the edges of a given uncolored graphGwith Δ + 1 colors. For this problem, our first parallel algorithm requiresO(Δ^5.5log³Δ logn+ Δ⁵log⁴n) time andO(max{n²Δ,nΔ³}) processors, which is a slight improvement on the algorithm by H. J. Karloff and D. B. Shmoys [J. Algorithms8 (1987), 39–52]. Their algorithm costsO(Δ⁶log⁴n) time andO(n²Δ) processors if we use the fastest known algorithm for finding maximal independent sets by M. Goldberg and T. Spencer [SIAM J. Discrete Math.2 (1989), 322–328]. Our second algorithm requiresO(Δ^4.5log³Δ logn+ Δ⁴log⁴n) time andO(max{n²,nΔ³}) processors. Finally, we present our third algorithm by incorporating the second algorithm as a subroutine. This algorithm requiresO(Δ^3.5log³Δ logn+ Δ³log⁴n) time andO(max{n²log Δ,nΔ³}) processors, which improves, by anO(Δ^2.5) factor in time, on Karloff and Shmoys' algorithm. All of these algorithms run in the COMMON CRCW PRAM model.     

An optimal parallel algorithm for computing cut vertices and blocks on interval graphs

In this paper, a parallel algorithm is presented to find all cut-vertices and blocks of an interval graph. If the list of sorted end points of the intervals of an interval graph is given then the proposed algorithm takes O(log n) time and O(n/log n) processors on an EREW PRAM, if the sorted list is not given then the time and processors complexities are respectively O(log n) and O(n).     

On a Nearest-Neighbor Problem Under Minkowski and Power Metrics for Large Data Sets

The paper presents the sequential and the parallel algorithm for solving the nearest-neighbor problem in the plane, based on the generalized Voronoi diagram construction. The applications of the problem are found in the areas of networking, communications, distributed systems, computer modeling and information retrieval. The input for the problem is the set of circular sites S with varying radii, the query point p and the metric (Minkowski or power) according to which the site, neighboring the query point, is to be reported. The sequential algorithm takes O(n) time to build the data structure and O(log n) time for each query. The parallel algorithm requires O(log n log) preprocessing time and O(log) query time on EREW PRAM architecture with n/log n processors. The IDG/NNM software was developed for an experimental study of the problem. The experimental results demonstrate that the Voronoi diagram method outperforms the k – d tree method for all tested input configurations. The tests were conducted on large data sets, comprising thousands of generators.     

Scalable 2D Convex Hull and Triangulation Algorithms for Coarse Grained Multicomputers

In this paper we describe scalable parallel algorithms for building the convex hull and a triangulation ofncoplanar points. These algorithms are designed for thecoarse grained multicomputermodel:pprocessors withO(n/p)⪢O(1) local memory each, connected to some arbitrary interconnection network. They scale over a large range of values ofnandp, assuming only thatn⩾p^1+ε(ε>0) and require timeO((T_sequential/p)+T_s(n, p)), whereT_s(n, p) refers to the time of a global sort ofndata on approcessor machine. Furthermore, they involve only a constant number of global communication rounds. Since computing either 2D convex hull or triangulation requires timeT_sequential=Θ(n log n) these algorithms either run in optimal time,Θ((n log n)/p), or in sort time,T_s(n, p), for the interconnection network in question. These results become optimal whenT_sequential/pdominatesT_s(n, p) or for interconnection networks like the mesh for which optimal sorting algorithms exist.     

Tridiagonalization of a symmetric matrix on a square array of mesh-connected processors

A parallel algorithm for transforming an n × n symmetric matrix to tridiagonal form is described. The algorithm implements Givens rotations on a square array of n × n processors in such a way that the transformation can be performed in time O(n log n). The processors require only nearest-neighbor communication. The reduction to tridiagonal form could be the first step in the parallel solution of the symmetric eigenvalue problem in time O(n log n).     

Efficient parallel solutions to some geometric problems

This paper presents new algorithms for solving some geometric problems on a shared memory parallel computer, where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location. The algorithms are quite different from known sequential algorithms, and are based on the use of a new parallel divide-and-conquer technique. One of our results is an O(log n) time, O(n) processor algorithm for the convex hull problem. Another result is an O(log n log log n) time, O(n) processor algorithm for the problem of selecting a closest pair of points among n input points.     

Parallel integer sorting and simulation amongst CRCW models

In this paper a general technique for reducing processors in simulation without any increase in time is described. This results in an O(√logn) time algorithm for simulating one step of PRIORITY on TOLERANT with processor-time product of O(n log logn); the same as that for simulating PRIORITY on ARBITRARY. This is used to obtain anO(logn/log logn + √logn (log logm ? log logn)) time algorithm for sortingn integers from the set {0,...,m ? 1},m ≧n, with a processor-time product ofO(n log logm log logn) on a TOLERANT CRCW PRAM. New upper and lower bounds for ordered chaining problem on an allocated COMMON CRCW model are also obtained. The algorithm for ordered chaining takesO(logn/log logn) time on an allocated PRAM of sizen. It is shown that this result is best possible (upto a constant multiplicative factor) by obtaining a lower bound of Ω(r logn/(logr + log logn)) for finding the first (leftmost one) live processor on an allocated-COMMON PRAM of sizen ofr-slow virtual processors (one processor simulatesr processors of allocated PRAM). As a result, for ordered chaining problem, “processor-time product” has to be at least Ω(n logn/log logn) for any poly-logarithmic time algorithm. Algorithm for ordered-chaining problem results in anO(logN/log logN) time algorithm for (stable) sorting ofn integers from the set {0,...,m ? 1} withn-processors on a COMMON CRCW PRAM; hereN = max(n, m). In particular if,m =n ^O(1), then sorting takes Θ(logn/log logn) time on both TOLERANT and COMMON CRCW PRAMs. Processor-time product for TOLERANT isO(n(log logn)²). Algorithm for COMMON usesn processors.     

Randomized parallel list ranking for distributed memory multiprocessors

We present a randomized parallel list ranking algorithm for distributed memory multiprocessors, using a BSP type model. We first describe a simple version which requires, with high probability, log(3p)+log ln(n)=Õ(logp+log logn) communication rounds (h-relations withh=Õ(n/p)) andÕ(n/p)) local computation. We then outline an improved version that requires high probability, onlyr?(4k+6) log(2/3p)+8=Õ(k logp) communication rounds wherek=min{i?0 |ln(i+1)n?(2/3p)²ⁱ⁺¹}. Notekn) is an extremely small number. Forn andp?4, the value ofk is at most 2. Hence, for a given number of processors,p, the number of communication rounds required is, for all practical purposes, independent ofn. Forn?1, 500,000 and 4?p?2048, the number of communication rounds in our algorithm is bounded, with high probability, by 78, but the actual number of communication rounds observed so far is 25 in the worst case. Forn?10010100 and 4?p?2048, the number of communication rounds in our algorithm is bounded, with high probability, by 118; and we conjecture that the actual number of communication rounds required will not exceed 50. Our algorithm has a considerably smaller member of communication rounds than the list ranking algorithm used in Reid-Miller’s empirical study of parallel list ranking on the Cray C-90.⁽¹⁾ To our knowledge, Reid-Miller’s algorithm⁽¹⁾ was the fastest list ranking implementation so far. Therefore, we expect that our result will have considerable practical relevance.