Parallel geometric algorithms on a mesh-connected computer

We show that a number of geometric problems can be solved on a n × n mesh-connected computer (MCC) inO(n) time, which is optimal to within a constant factor, since a nontrivial data movement on an MCC requires (n) time. The problems studied here include multipoint location, planar point location, trapezoidal decomposition, intersection detection, intersection of two convex polygons, Voronoi diagram, the largest empty circle, the smallest enclosing circle, etc. TheO(n) algorithms for all of the above problems are based on the classical divide-and-conquer problem-solving strategy.This work was supported in part by the National Science Foundation under Grant DCR 8420814. A preliminary version was presented in the 1987 FJCC, Dallas, TX.     

Parallel geometric algorithms on a mesh-connected computer

We show that a number of geometric problems can be solved on a √n × √n mesh-connected computer (MCC) inO(√n) time, which is optimal to within a constant factor, since a nontrivial data movement on an MCC requires Ω(√n) time. The problems studied here include multipoint location, planar point location, trapezoidal decomposition, intersection detection, intersection of two convex polygons, Voronoi diagram, the largest empty circle, the smallest enclosing circle, etc. TheO(√n) algorithms for all of the above problems are based on the classical divide-and-conquer problem-solving strategy.     

Geometric algorithms for digitized pictures on a mesh-connected computer

Although mesh-connected computers are used almost exclusively for low-level local image processing, they are also suitable for higher level image processing tasks. We illustrate this by presenting new optimal (in the O-notational sense) algorithms for computing several geometric properties of figures. For example, given a black/white picture stored one pixel per processing element in an n × n mesh-connected computer, we give ?(n) time algorithms for determining the extreme points of the convex hull of each component, for deciding if the convex hull of each component contains pixels that are not members of the component, for deciding if two sets of processors are linearly separable, for deciding if each component is convex, for determining the distance to the nearest neighboring component of each component, for determining internal distances in each component, for counting and marking minimal internal paths in each component, for computing the external diameter of each component, for solving the largest empty circle problem, for determining internal diameters of components without holes, and for solving the all-points farthest point problem. Previous mesh-connected computer algorithms for these problems were either nonexistent or had worst case times of ?(n2). Since any serial computer has a best case time of ?(n2) when processing an n × n image, our algorithms show that the mesh-connected computer provides significantly better solutions to these problems.     

Two packet routing algorithms on a mesh-connected computer

We give two algorithms for the 1-1 routing problems on a mesh-connected computer. The first algorithm, with queue size 28, solves the 1-1 routing problem on an n×n mesh-connected computer in 2n+O(1) steps. This improves the previous queue size of 75. The second algorithm solves the 1-1 routing problem in 2n-2 steps with queue size 12 t_s/s where t_s is the time for sorting an s×s mesh into a row major order for all s⩾1. This result improves the previous queue size 18.67 t_s/s     

Optimal routing algorithms for mesh-connected processor arrays

We show that there is a randomizedoblivious algorithm for routing any (partial) permutation on ann ×n grid in 2n +O(logn) parallel communication steps. The queues will not grow larger than Θ(logn/log logn) with high probability. We then modify this to obtain a (nonoblivious) algorithm with the same running time such that the size of the queues is bounded by a constant with high probability. For permutations withlocality, where each packet has to travel a distance at mostL, a generalization of the algorithm routes in time proportional toL with high probability. Finally, we identify a class of meshlike networks that have optimal or near-optimal diameter. These meshes have the potential of being adapted to run existing sorting and routing algorithms with corresponding reduction in their running times.     

Optimal routing algorithms for mesh-connected processor arrays

We show that there is a randomizedoblivious algorithm for routing any (partial) permutation on ann ×n grid in 2n +O(logn) parallel communication steps. The queues will not grow larger than (logn/log logn) with high probability. We then modify this to obtain a (nonoblivious) algorithm with the same running time such that the size of the queues is bounded by a constant with high probability. For permutations withlocality, where each packet has to travel a distance at mostL, a generalization of the algorithm routes in time proportional toL with high probability. Finally, we identify a class of meshlike networks that have optimal or near-optimal diameter. These meshes have the potential of being adapted to run existing sorting and routing algorithms with corresponding reduction in their running times.Preliminary reports of portions of the results contained in this paper have appeared in theProceedings of the 1988 Aegean Workshop on Computing [5], and in theProceedings of the 1987 Conference on Foundations of Software Technology and Theoretical Computer Science [18]. The work of the first author was supported in part by NSF Grant NSF-DCR-85-03251 and ONR contract N00014-80-C-0647. The work of the second author was supported in part by NSF Grant NSF-DCR-86-00379.     

Optimal algorithm for mutual exclusion in mesh-connected computer networks

A distributed algorithm is presented that realizes mutual exclusion among n nodes in a mesh-connected computer network. The nodes communicate by using messages only, and there is no global controller. The algorithm requires at most 3.5 √n messages per mutual exclusion invocation under light demand, and reduces to approximately four messages under heavy demand. The time required to achieve mutual exclusion is shown to be minimal under some general assumptions.     

Optical clustering on a mesh-connected computer

In this paper, we present optimal parallel algorithms for optical clustering on a mesh-connected computer.Optical clustering is a clustering technique based on the principal of optical resolution, and is of particular interest in picture analysis. The algorithms we present are based on the application of parallel algorithms in computational geometry and graph theory. In particular, we show that given a setS ofN points in the Euclidean plane, the following problems can be solved in optimal time on a mesh-connected computer of sizeN.

Space and time optimal algorithms for a class of rectangle intersection problems

The intersection problem for a subclass of rectangles called r-rectangles is investigated and reduced to the balanced batched r(estricted)-range searching problem as well as to the balanced batched inverse r-range searching problem. Simple algorithms for these problems are given which are space and time optimal. The algorithm given for the balanced batched r-range searching problem leads to a new algorithm for the all-points ECDF problem in 2-space which is simple and optimal. Again, the balanced batched r-range searching algorithm is combined with a known algorithm for batched range searching problems, leading to a new algorithm for the rectangle intersection problem which is space and time optimal in the worst case when the given set of rectangles contains a much higher proportion of r-rectangles.     

Optimal architectures and algorithms for mesh-connected parallelcomputers with separable row/column buses

A two-dimensional mesh of processing elements (PE's) with separable row and column buses (i.e., broadcast mechanisms for rows and columns that can be logically divided into a number of local buses through the use of PE-controlled switches) has been shown to be quite effective for semigroup computation, prefix computation, and a wide class of other computations that do not require excessive communication or data routing. For meshes with separable row/column buses, the authors show how semigroup and prefix computations can be performed with the same asymptotic time complexity without the provision of buses for every row and every column and discuss the VLSI implications of this new architecture     

Parallel reconfiguration algorithms for mesh-connected processor arrays

Effective fault tolerance techniques are essential for improving the reliability of multiprocessor systems. At the same time, fault tolerance must be achieved at high speed to meet the real-time constraints of embedded systems. While parallelism has often been exploited to increase performance, to the best of our knowledge, there has been no previously reported work on parallel reconfiguration of mesh-connected processor arrays with faults. This paper presents two parallel algorithms to accelerate reconfiguration of the processor arrays. The first algorithm reconfigures a host array in parallel in a multithreading manner. The threads in the parallel algorithm execute independently within a safe rerouting distance. The second algorithm is based on a divide-and-conquer approach to first generate the leftmost segments in parallel and then merge the segments in parallel. When compared to the conventional algorithm, simulation results from a large number of instances confirm that the proposed algorithms significantly accelerate the reconfiguration without loss of harvest.     

Prefix computations on a generalized mesh-connected computer withmultiple buses

The mesh-connected computer with multiple buses (MC-CMB) is a well-known parallel organization, providing broadcast facilities in each row and each column. In this paper, we propose a 2D generalized MCCMB (2-GMCCMB) for the purpose of increasing the efficiency of executing some important applications of prefix computations such as solving Linear recurrences and tridiagonaI systems, etc. A k₁n₁ ×k₁n₂ 2-GMCCMB is constructed from a k ₁n₁×k₁n₂ mesh organization by enhancing the power of each disjoint n₁×n₂ submesh with multiple buses (sub-2-MCCMB). Given n data, a prefix computation can be performed in O(n^1/10) time on an n^3/5×n^2/5 2-GMCCMB, where each disjoint sub-2-MCCMB is of size n^1/2×n^3/10. This time bound is faster than the previous time bound of O(n^1/8) for the same computation on an n^5/8×n^3/8 2-MCCMB. Furthermore, the time bound of our parallel prefix algorithm can be further reduced to O(n^1/11) if fewer processors are used. Our result can be extended to the d-dimensional GMCCMB, giving a time bound of O(n¹ (d2(d)+d)/) for any constant d; here, we omit the constant factors. This time bound is less than the previous time bound of O(n¹(d2(d))/) on the d-dimensional MCCMB     

Hough transform algorithms for mesh-connected SIMD parallel processors

Hough transform techniques for straight line detection play a key role in the road following algorithms developed by the University of Maryland for the DARPA Autonomous Land Vehicle Project. This report compares several methods of Hough transform computation suitable for implementation on a mesh-connected SIMD parallel processor, such as Goddard Space Flight Center's Massively Parallel Processor (MPP) or Martin Marietta Corp.'s Geometric Arithmetic Parallel Processor (GAPP).     

Optimal algorithms for some intersection radius problems

The intersection radius of a set ofn geometrical objects in ad-dimensional Euclidean space,E ^d , is the radius of the smallest closed hypersphere that intersects all the objects of the set. In this paper, we describe optimal algorithms for some intersection radius problems. We first present a linear-time algorithm to determine the smallest closed hypersphere that intersects a set of hyperplanes inE ^d , assumingd to be a fixed parameter. This is done by reducing the problem to a linear programming problem in a (d+1)-dimensional space, involving 2n linear constraints. We also show how the prune-and-search technique, coupled with the strategy of replacing a ray by a point or a line can be used to solve, in linear time, the intersection radius problem for a set ofn line segments in the plane. Currently, no algorithms are known that solve these intersection radius problems within the same time bounds.     

Optimal parallel algorithms for rectilinear link-distance problems

We provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). LetP be a trapezoided rectilinear simple polygon withn vertices. InO(logn) time usingO(n/logn) processors we can optimally compute:

1. Minimum réctilinear link paths, or shortest paths in theL ₁ metric from any point inP to all vertices ofP.
2. Minimum rectilinear link paths from any segment insideP to all vertices ofP.
3. The rectilinear window (histogram) partition ofP.
4. Both covering radii and vertex intervals for any diagonal ofP.
5. A data structure to support rectilinear link-distance queries between any two points inP (queries can be answered optimally inO(logn) time by uniprocessor).

Our solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which usedO(n logn) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon.     

Optimal parallel algorithms for problems modeled by a family ofintervals

A family of intervals on the real line provides a natural model for a vast number of scheduling and VLSI problems. Recently, a number of parallel algorithms to solve a variety of practical problems on such a family of intervals have been proposed in the literature. The authors develop computational tools and show how they can be used for the purpose of devising cost-optimal parallel algorithms for a number of interval-related problems, including finding a largest subset of pairwise nonoverlapping intervals, a minimum dominating subset of intervals, along with algorithms to compute the shortest path between a pair of intervals and, based on the shortest path, a parallel algorithm to find the center of the family of intervals. More precisely, with an arbitrary family of n intervals as input, all the algorithms run in O(log n) time using O(n) processors in the EREW-PRAM model of computation     

Propagation algorithms for framing rectangle construction

Given a binary image containing a connected component, parallel propagation algorithms are presented for constructing upright and 45°-tilted framing rectangles around the component in time proportional to the sum of the dimensions of these rectangles. The algorithms make use of properties of geodesics connecting pairs of points in the component to iteratively fill in the region to the desired shape.     

Processor-time optimal parallel algorithms for digitized images on mesh-connected processor arrays

We present processor-time optimal parallel algorithms for several problems onn ×n digitized image arrays, on a mesh-connected array havingp processors and a memory of sizeO(n ²) words. The number of processorsp can vary over the range [1,n ^3/2] while providing optimal speedup for these problems. The class of image problems considered here includes labeling the connected components of an image; computing the convex hull, the diameter, and a smallest enclosing box of each component; and computing all closest neighbors. Such problems arise in medium-level vision and require global operations on image pixels. To achieve optimal performance, several efficient data-movement and reduction techniques are developed for the proposed organization.     

Approximation algorithms for partitioning a rectangle with interior points

LetR be a rectangle and letP be a set of points located insideR. Our problem consists of introducing a set of line segments of least total length to partition the interior ofR into rectangles. Each rectangle in a valid partition must not contain points fromP as interior points. Since this partitioning problem is computationally intractable (NP-hard), we present efficient approximation algorithms for its solution. The solutions generated by our algorithms are guaranteed to be within three times the optimal solution value. Our algorithm also generates solutions within four times the optimal solution value whenR is a rectilinear polygon. Our algorithm can be generalized to generate good approximation solutions for the case whenR is a rectilinear polygon, there are rectilinear polygonal holes, and the sum of the length of the boundaries is not more than the sum of the length of the edges in an optimal solution.