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28
Towards Optimal Locality in MeshIndexings
, 1997
"... The efficiency of many data structures and algorithms relies on "localitypreserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slowgrowing function of ji jj. We present a new 2D indexing scheme we call H ..."
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Cited by 31 (4 self)
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The efficiency of many data structures and algorithms relies on "localitypreserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slowgrowing function of ji jj. We present a new 2D indexing scheme we call Hindexing , which has superior (possibly optimal) locality in comparison with the wellknown Hilbert indexings. Hindexings form a Hamiltonian cycle and we prove that they are optimally localitypreserving among all cyclic indexings. We provide fairly tight lower bounds for indexings without any restriction. Finally, illustrated by investigations concerning 2D and 3D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.
Constant Time Algorithms for Computational Geometry on the Reconfigurable Mesh
 IEEE Transactions on Parallel and Distributed Systems
, 1997
"... The reconfigurable mesh consists of an array of processors interconnected by a reconfigurable bus system. The bus system can be used to dynamically obtain various interconnection patterns among the processors. Recently, this model has attracted a lot of attention. In this paper, we show O(1) time so ..."
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Cited by 18 (2 self)
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The reconfigurable mesh consists of an array of processors interconnected by a reconfigurable bus system. The bus system can be used to dynamically obtain various interconnection patterns among the processors. Recently, this model has attracted a lot of attention. In this paper, we show O(1) time solutions to the following computational geometry problems on the reconfigurable mesh: allpairs nearest neighbors, convex hull, triangulation, twodimensional maxima, twoset dominance counting, and smallest enclosing box. All these solutions accept N planar points as input and employ an N  N reconfigurable mesh. The basic scheme employed in our implementations is to recursively find an O(1) time solution. The number of recursion levels and the size of the subproblems at each level of recursion are optimized such that the problem decomposition and the solution to the problem can be obtained in constant time. As a result, we have developed some efficient merge techniques to combine th...
Parallel techniques for computational geometry
 Proc. IEEE
, 1992
"... A survey of techniques for solving geometric problems in parallel is given, both for shared memory parallel machines and for networks of processors. Open problems are also discussed, as well as directions for future research. 'This work was supported by the office oi Naval Research under Contracts N ..."
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Cited by 13 (0 self)
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A survey of techniques for solving geometric problems in parallel is given, both for shared memory parallel machines and for networks of processors. Open problems are also discussed, as well as directions for future research. 'This work was supported by the office oi Naval Research under Contracts N0001484K0502 and
Square Meshes Are Not Optimal For Convex Hull Computation
 IEEE Transactions on Parallel and Distributed Systems
"... Recently it has been noticed that for semigroup computations and for selection rectangular meshes with multiple broadcasting yield faster algorithms than their square counterparts. The contribution of this paper is to provide yet another example of a fundamental problem for which this phenomenon ..."
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Cited by 12 (9 self)
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Recently it has been noticed that for semigroup computations and for selection rectangular meshes with multiple broadcasting yield faster algorithms than their square counterparts. The contribution of this paper is to provide yet another example of a fundamental problem for which this phenomenon occurs. Specifically, we show that the problem of computing the convex hull of a set of n sorted points in the plane can be solved in O(n 1 8 log 3 4 n) time on a rectangular mesh with multiple broadcasting of size n 3 8 log 1 4 n \Theta n 5 8 log 1 4 n . The fastest previouslyknown algorithms on a square mesh of size p n \Theta p n run in O(n 1 6 ) time in case the n points are pixels in a binary image, and in O(n 1 6 log 2 3 n) time for sorted points in the plane. Keywords: convex hulls, meshes with broadcasting, parallel algorithms, pattern recognition, image processing, computational geometry. 1 Introduction One of the fundamental heuristics in pat...
Digital Analog Simulation Of Uniform Motion In Representations Of Physical NSpace By LatticeWork MIMD Computer Architectures
, 1991
"... This doctoral dissertation is part of an ongoing research project with John Case, Dayanand S. Rajan and myself. We are investigating the possibility of solving problems in scientific computing involving the motion of objects in a bounded region of physical nspace by (a) representing points in the r ..."
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Cited by 8 (7 self)
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This doctoral dissertation is part of an ongoing research project with John Case, Dayanand S. Rajan and myself. We are investigating the possibility of solving problems in scientific computing involving the motion of objects in a bounded region of physical nspace by (a) representing points in the region of space by processors in a latticework mesh of processors with local connections for interprocessor communication, and (b) literally, analogically simulating the motion of objects by representing the particles of these objects by algorithms which move themselves about in the latticework of processors, much as the motion in real space of the particles making up real objects, in effect, constitutes the motion of those objects. The main contributions of this dissertation are (i) two provably correct algorithms to generate virtually perfectly shaped spherical wavefronts emanating from a point source at virtually constant radial speed, (ii) a provably correct algorithm template for simu...
Multisearch Techniques: Parallel Data Structures on MeshConnected Computers
 Journal of Parallel and Distributed Computing
, 1994
"... The {\em multisearch problem} is defined as follows. Given a data structure $D$ modeled as a graph with $n$ constantdegree nodes, perform $O(n)$ searches on $D$. Let $r$ be the length of the longest search path associated with a search process, and assume that the paths are determined ``online''. ..."
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Cited by 7 (2 self)
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The {\em multisearch problem} is defined as follows. Given a data structure $D$ modeled as a graph with $n$ constantdegree nodes, perform $O(n)$ searches on $D$. Let $r$ be the length of the longest search path associated with a search process, and assume that the paths are determined ``online''. That is, the search paths may overlap arbitrarily. In this paper, we solve the multisearch problem for certain classes of graphs in $O(\sqrt{n} + {r} \frac{\sqrt{n}}{\log n})$ time on a $\sqrt{n} \times \sqrt{n}$ meshconnected computer. For many data structures, the search path traversed when answering one search query has length $r=O(\log n)$. For these cases, our algorithm processes $O(n)$ such queries in asymptotically optimal $\Theta(\sqrt{n})$ time. The classes of graphs we consider contain many of the important data structures that arise in practice, ranging from simple trees to Kirkpatrick hierarchical search DAGs. Multisearch is a useful abstraction that can be used to implement parallel versions of standard sequential data structures on a mesh. As example applications, we consider a variety of parallel online tree traversals, as well as hierarchical representations of polyhedra and its myriad of applications (linespolyhedron intersection queries, multiple tangent plane determination, intersecting convex polyhedra, and threedimensional convex hull).
Complexity of SubBus Mesh Computations
 Dept. of CS&EE, U. of Washington
, 1996
"... . The time complexity of several fundamental problems on the subbus mesh parallel computer with p processors is investigated. The problems include computing the PARITY and MAJORITY of p bits, the SUM of p numbers of length O(logp) and the MINIMUM of p numbers. It is shown that in one dimension the ..."
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Cited by 4 (1 self)
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. The time complexity of several fundamental problems on the subbus mesh parallel computer with p processors is investigated. The problems include computing the PARITY and MAJORITY of p bits, the SUM of p numbers of length O(logp) and the MINIMUM of p numbers. It is shown that in one dimension the time to compute any of these problems is 2(log p). In two dimensions the time to compute any of PARITY, MAJORITY, and SUM is 2( log p log log p ). It was previously shown that the time to compute MINIMUM in two dimensions is 2(log log p) [0, 0]. Key words. subbus mesh, reconfigurable mesh, time complexity, parity, majority, sum, minimum AMS subject classifications. 68Q10, 68Q15, 68Q22, 68Q25 1. Introduction. A subbus mesh computer is a singleinstruction multipledata (SIMD) twodimensional array of processors, where processors can broadcast data vertically or horizontally on segmented busses. On a segmented bus, some of the processors on the bus are active while others are inactive....
TimeOptimal NearestNeighbor Computations on Enhanced Meshes
 Proc. PARLE
, 1994
"... The AllNearest Neighbor problem (ANN, for short) is stated as follows: given a set S of points in the plane, determine for every point in S, a point that lies closest to it. The ANN problem is central to VLSI design, computer graphics, pattern recognition, and image processing, among others. In th ..."
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Cited by 4 (2 self)
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The AllNearest Neighbor problem (ANN, for short) is stated as follows: given a set S of points in the plane, determine for every point in S, a point that lies closest to it. The ANN problem is central to VLSI design, computer graphics, pattern recognition, and image processing, among others. In this paper we propose timeoptimal algorithms to solve the ANN problem for an arbitrary set of points in the plane and also for the special case in which the points are vertices of a convex polygon. Both our algorithms run on meshes with multiple broadcasting. We first establish an\Omega\Gamma/30 n) time lower bound for the task of solving an arbitrary npoint instance of the ANN problem, even if the points are the vertices of a convex polygon. This lower bound holds for both the CREWPRAM and for the mesh with multiple broadcasting. Next, we show that the bound is tight by exhibiting algorithms solving the problem in O(log n) time on a mesh with multiple broadcasting of size n \Theta n. Keyw...
Optimal Pattern Matching on Meshes
 Proc. 11th Symposium on Theoretical Aspects of Computer Science
, 1993
"... . Parallel pattern matching on a meshconnected array of processors is considered. The problem is to find all occurrences of a pattern in a text. The input text is a string of n symbols placed in a p n \Theta p n mesh, each processor storing one symbol. The pattern is stored similarly in a conti ..."
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Cited by 4 (2 self)
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. Parallel pattern matching on a meshconnected array of processors is considered. The problem is to find all occurrences of a pattern in a text. The input text is a string of n symbols placed in a p n \Theta p n mesh, each processor storing one symbol. The pattern is stored similarly in a contiguous portion of the mesh. An algorithm solving the problem in time O( p n) is presented. It applies a novel technique to design parallel patternmatching algorithms based on the notion of a pseudoperiod. 1 Introduction The problem of pattern matching is to find all occurrences of a given pattern in a given text. A parallel algorithm solving this problem on a meshconnected computer is presented. This parallel computer is a p n \Theta p n array of n processors interconnected according to a grid pattern. Suppose a text t is a string of n symbols taken from some alphabet, and a pattern p is a string of m symbols, m n, from the same alphabet (no restrictions on the size of alphabets are...
The Mesh with Hybrid Buses: An Efficient Parallel Architecture for Digital Geometry
, 1999
"... The first main contribution of this work is to propose an efficient VLSI architecture obtained by augmenting the Mesh with Multiple Broadcasting (MMB) with precharged 1bit row and column buses. The new architecture that we call Mesh with Hybrid Buses (MHB, for short) is realizable in VLSI with no i ..."
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Cited by 4 (1 self)
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The first main contribution of this work is to propose an efficient VLSI architecture obtained by augmenting the Mesh with Multiple Broadcasting (MMB) with precharged 1bit row and column buses. The new architecture that we call Mesh with Hybrid Buses (MHB, for short) is realizable in VLSI with no increase in the area or the wiring complexity of the MMB chip. Our second main contribution is to show that the MHB is extremely well suited for solving an entire slew of digital geometry tasks. The MHB is not a reconfigurable architecture. Yet, quite remarkably, for a large number of fundamental digital geometry tasks, the MHB offers a level of performance previously attained only by reconfigurable architectures. Specifically, with a digital image pretiled onto a MHB of size p n \Theta p n one pixel per processor, we show that the problems of: computing the convex hull of the image, computing the diameter and the width of the image, deciding whether a set of digital points is a digital li...