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37
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 ..."
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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 19 (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, 80:1435– 1448
, 1992
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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&apo ..."
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Cited by 8 (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).
On the ManhattanDistance Between Points on SpaceFilling MeshIndexings
, 1996
"... Indexing schemes based on space filling curves like the Hilbert curve are a powerful tool for building efficient parallel algorithms on meshconnected computers. The main reason is that they are localitypreserving, i.e., the Manhattandistance between processors grows only slowly with increasing in ..."
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Cited by 5 (0 self)
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Indexing schemes based on space filling curves like the Hilbert curve are a powerful tool for building efficient parallel algorithms on meshconnected computers. The main reason is that they are localitypreserving, i.e., the Manhattandistance between processors grows only slowly with increasing index differences. We present a simple and easytoverify proof that the Manhattandistance of any indices i and j is bounded by 3 p ji \Gamma jj \Gamma 2 for the 2DHilbert curve. The technique used for the proof is then generalized for a large class of selfsimilar curves. We use this result to show a (quite tight) bound of 4:73458 3 p ji \Gamma jj \Gamma 3 for a 3DHilbert curve. 1 Introduction It has become increasingly clear that meshconnected processor arrays, grids for short, are among the most realistic models of parallel computation [1, 4, 14, 18]. The indexing of the processors is an important aspect in the design of mesh algorithms. Several indexing schemes are wellknown. Mos...
A sliding memory plane array processor for low level vision
 in Proc. Int. Conj Pattern Recognition, Atlantic
, 1990
"... AbstractThis paper describes a new meshconnected SIMD ..."
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AbstractThis paper describes a new meshconnected SIMD
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....
Spatial/Kinematic Domain and Lattice Computers
 JOURNAL OF EXPERIMENTAL AND THEORETICAL ARTIFICIAL INTELLIGENCE
, 1994
"... An approach to analogical representation for objects and their motions in space is proposed. This approach involves lattice computer architectures and associated algorithms and is shown to be abstracted from the behavior of human beings mentally solving spatial /kinematic puzzles. There is also dis ..."
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Cited by 4 (4 self)
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An approach to analogical representation for objects and their motions in space is proposed. This approach involves lattice computer architectures and associated algorithms and is shown to be abstracted from the behavior of human beings mentally solving spatial /kinematic puzzles. There is also discussion of where in this approach the modeling of human cognition leaves off and the engineering begins. The possible relevance of the approach to a number of issues in Artificial Intelligence is discussed. These issues include efficiency of sentential versus analogical representations, common sense reasoning, update propagation, learning performance tasks, diagrammatic representations, spatial reasoning, metaphor, human categorization, and pattern recognition. Lastly there is a discussion of the somewhat related approach involving cellular automata applied to computational physics.