Mesh Computer Algorithms for Computational Geometry

Asymptotically optimal parallel algorithms are presented for use on a mesh computer to determine several fundamental geometric properties of figures. For example, given multiple figures represented by the Cartesian coordinates of n or fewer planar vertices, distributed one point per processor on a two-dimensional mesh computer with n simple processing elements, Theta (n/sup 1/2/>or=-time algorithms are given for identifying the convex hull and smallest enclosing box of each figure. Given two such figures, a Theta (n/sup 1/2/>or=-time algorithm is given to decide if the two figures are linearly separable. Given n or fewer planar points, Theta (n/sup 1/2/>or=-time algorithms are given to solve the all-nearest neighbor problems for points and for sets of points. Given n or fewer circles, convex figures, hyperplanes, simple polygons, orthogonal polygons, or iso-oriented rectangles, Theta (n/sup 1/2/>or=-time algorithms are given to solve a variety of area and intersection problems. Since any serial computer has worst-case time of Omega (n) when processing n points, these algorithms show that the mesh computer provides significantly better solutions to these problems. >

[1]  Bruce W. Weide,et al.  Optimal Expected-Time Algorithms for Closest Point Problems , 1980, TOMS.

[2]  Davis Avis,et al.  On the complexity of finding the convex hull of a set of points , 1982, Discret. Appl. Math..

[3]  Mikhail J. Atallah,et al.  Solving Tree Problems on a Mesh-Connected Processor Array (Preliminary Version) , 1985, FOCS.

[4]  H. T. Kung,et al.  Sorting on a mesh-connected parallel computer , 1977, CACM.

[5]  Frans C. A. Groen,et al.  The smallest box around a package , 1981, Pattern Recognit..

[6]  Godfried T. Toussaint,et al.  PATTERN RECOGNITION AND GEOMETRICAL COMPLEXITY. , 1980 .

[7]  Frank Dehne,et al.  O(n^(1/2)) Algorithms for the Maximal Elements and ECDF Searching Problem on a Mesh-Connected Parallel Computer , 1986, Inf. Process. Lett..

[8]  Azriel Rosenfeld,et al.  Parallel Image Processing by Memory-Augmented Cellular Automata , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Kenneth E. Batcher,et al.  Design of a Massively Parallel Processor , 1980, IEEE Transactions on Computers.

[10]  JOSEPH O’ROURKE,et al.  A new linear algorithm for intersecting convex polygons , 1982, Comput. Graph. Image Process..

[11]  Lynn Conway,et al.  Introduction to VLSI systems , 1978 .

[12]  Leonidas J. Guibas,et al.  Parallel computational geometry , 1988, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[13]  Quentin F. Stout,et al.  Mesh-Connected Computers with Broadcasting , 1983, IEEE Transactions on Computers.

[14]  Michael Ian Shamos,et al.  Geometric intersection problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[15]  Mikhail J. Atallah,et al.  Graph Problems on a Mesh-Connected Processor Array , 1984, JACM.

[16]  D. T. Lee,et al.  Computational Geometry—A Survey , 1984, IEEE Transactions on Computers.

[17]  Richard M. Brown,et al.  The ILLIAC IV Computer , 1968, IEEE Transactions on Computers.

[18]  Herbert Freeman,et al.  Determining the minimum-area encasing rectangle for an arbitrary closed curve , 1975, CACM.

[19]  S. N. Maheshwari,et al.  Parallel algorithms for the convex hull problem in two dimensions , 1981, CONPAR.

[20]  Quentin F. Stout.  Topological matching , 1983, STOC '83.

[21]  Bernard Chazelle.  Computational Geometry on a Systolic Chip , 1984, IEEE Transactions on Computers.

[22]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[23]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[24]  Robert E. Tarjan,et al.  Application of a Planar Separator Theorem , 1977, FOCS.

[25]  Sartaj Sahni,et al.  Data broadcasting in SIMD computers , 1981, IEEE Transactions on Computers.

[26]  Russ Miller,et al.  Varying Diameter and Problem Size in Mesh-Connected Computers , 1985, ICPP.

[27]  Quentin F. Stout,et al.  BROADCASTING IN MESH-CONNECTED COMPUTERS. , 1982 .

[28]  Andrew Chi-Chih Yao,et al.  A Lower Bound to Finding Convex Hulls , 1981, JACM.

[29]  Quentin F. Stout.  Tree-Based Graph Algorithms for Some Parallel Computers , 1985, ICPP.

[30]  Kai Hwang,et al.  Integrated Computer Architectures for Image Processing and Database Management , 1983, Computer.

[31]  Jon Louis Bentley,et al.  Multidimensional divide-and-conquer , 1980, CACM.

[32]  Mi Lu.  Constructing the Voronoi Diagram on a Mesh-Connected Computer , 1986, ICPP.

[33]  Russ Miller,et al.  COMPUTATIONAL GEOMETRY ON A MESH-CONNECTED COMPUTER. , 1984 .

[34]  Ian Watson,et al.  A Practical Data Flow Computer , 1982, Computer.

[35]  Edward M. McCreight,et al.  Priority Search Trees , 1985, SIAM J. Comput..

[36]  D. M. Watson,et al.  The Cellular Logic Array Image Processor , 1977, Comput. J..

[37]  Russ Miller,et al.  Geometric Algorithms for Digitized Pictures on a Mesh-Connected Computer , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[38]  Mikhail J. Atallah,et al.  Efficient Parallel Solutions to Some Geometric Problems , 1986, J. Parallel Distributed Comput..

[39]  Sartaj Sahni,et al.  Optimal BPC Permutations on a Cube Connected SIMD Computer , 1982, IEEE Transactions on Computers.