## Constant Time Algorithms for Computational Geometry on the Reconfigurable Mesh (1997)

Venue: | IEEE Transactions on Parallel and Distributed Systems |

Citations: | 17 - 2 self |

### BibTeX

@ARTICLE{Jang97constanttime,

author = {Ju-wook Jang and Madhusudan Nigam and Viktor K. Prasanna and Sartaj Sahni},

title = {Constant Time Algorithms for Computational Geometry on the Reconfigurable Mesh},

journal = {IEEE Transactions on Parallel and Distributed Systems},

year = {1997},

volume = {8},

pages = {1--12}

}

### OpenURL

### Abstract

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: all-pairs nearest neighbors, convex hull, triangulation, two-dimensional maxima, two-set 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...

### Citations

1762 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...(i, j) have copies of the u(v)th point of E(Si)(E(Sj)), for 0 £ uv , < N. For all u, v, the (u, v)th PE of BM(i, j) computes the 1. An alternative to using the polar angle is discussed on page 100 of=-= [36]-=-.sJANG ET AL.: CONSTANT TIME ALGORITHMS FOR COMPUTATIONAL GEOMETRY ON THE RECONFIGURABLE MESH 5 slope of the line connecting the uth point of E(Si) and the vth point of E(Sj). Since the points of E(Sj... |

85 | The Power of Reconfiguration
- Ben-Asher
- 1991
(Show Context)
Citation Context ...n the theoretical side, many efficient parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], =-=[1]-=-, [29], [22], [46], selection [7], [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for comput... |

71 |
Polymorphic-Torus Network
- Li, Maresca
- 1989
(Show Context)
Citation Context ...e programs in execution. It shares some basic features with the CHiP computer [41], mesh connected computers augmented with broadcast buses [35], the bus automaton [39], the polymorphic-torus network =-=[19]-=-, and the coterie network in the latest version of the Content Addressable Array Parallel Processor (CAAPP) [50]. A reconfigurable mesh has been built by NEC which has 512 PEs [18]. Researchers at IBM... |

57 |
Constant time algorithms for the transitive closure and some related graph problems on processor arrays with reconfigurable bus systems
- Wang, Chen
- 1990
(Show Context)
Citation Context ... fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection [7], [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], =-=[47]. In this paper, we develop consta-=-nt time algorithms for computational geometry problems including convex hull, kdimensional maxima, two-set dominance counting, smallest ———————————————— • J. Jang is ... |

53 |
Meshes with reconfigurable buses
- Miller
- 1988
(Show Context)
Citation Context ...ms on the reconfigurable mesh. These techniques exploit reconfigurability in nontrivial ways leading to constant time solutions using optimal size of the mesh. 1 INTRODUCTION T HE reconfigurable mesh =-=[24]-=- is a variant of the mesh connected computer model in which the shape of the buses can be altered to suit the need of the programs in execution. It shares some basic features with the CHiP computer [4... |

51 |
An optimal sorting algorithm on reconfigurable mesh
- Jang, Prasanna
- 1995
(Show Context)
Citation Context ...ses. On the theoretical side, many efficient parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting =-=[10]-=-, [1], [29], [22], [46], selection [7], [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for c... |

42 |
Array Processor with Multiple Broadcasting
- Kumar, Raghavendra
- 1987
(Show Context)
Citation Context ...the shape of the buses can be altered to suit the need of the programs in execution. It shares some basic features with the CHiP computer [41], mesh connected computers augmented with broadcast buses =-=[35]-=-, the bus automaton [39], the polymorphic-torus network [19], and the coterie network in the latest version of the Content Addressable Array Parallel Processor (CAAPP) [50]. A reconfigurable mesh has ... |

39 |
Bus automata, brains, and mental models
- Rothstein
- 1988
(Show Context)
Citation Context ...an be altered to suit the need of the programs in execution. It shares some basic features with the CHiP computer [41], mesh connected computers augmented with broadcast buses [35], the bus automaton =-=[39]-=-, the polymorphic-torus network [19], and the coterie network in the latest version of the Content Addressable Array Parallel Processor (CAAPP) [50]. A reconfigurable mesh has been built by NEC which ... |

36 |
A Fast Algorithm for Computing Histograms on a Reconfigurable Mesh
- Jang, Park, et al.
- 1992
(Show Context)
Citation Context ...igurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection [7], [9], multiplication [13], division [34], histogramming =-=[12]-=-, [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry problems including convex hull, kdimensional maxima, two-set dominance counti... |

35 |
Introduction to the configurable, highly parallel computer
- Snyder
- 1982
(Show Context)
Citation Context ...4] is a variant of the mesh connected computer model in which the shape of the buses can be altered to suit the need of the programs in execution. It shares some basic features with the CHiP computer =-=[41]-=-, mesh connected computers augmented with broadcast buses [35], the bus automaton [39], the polymorphic-torus network [19], and the coterie network in the latest version of the Content Addressable Arr... |

34 |
Constant time sorting on a processor array with a reconfigurable bus system
- Wang, Chen, et al.
- 1990
(Show Context)
Citation Context ... side, many efficient parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], =-=[46]-=-, selection [7], [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry p... |

33 |
Mesh computer algorithms for computational geometry
- Miller, Stout
- 1989
(Show Context)
Citation Context ...set S of N planar points, the all-pairs nearest neighbor problem can be defined as follows: For each p ° S, find a q ° S, such that q › p and q is nearest to p among all points in S. Miller and St=-=out [25] -=-solve the problem in O( N) time using log N levels of recursion on a 2-MCC of size N ¥ N . In [23], Mackenzie and Stout solve it using log log N levels of recursion on a hypercube of size N to run in... |

23 | Meshes with multiple buses - Stout - 1986 |

21 |
Thiruchelvan, On the Power of Segmenting and Fusing Buses
- Trahan, Vaidyanathan, et al.
- 1996
(Show Context)
Citation Context ...esh wrap around (Fig. 5). A variant of the reconfigurable mesh model has been studied in [20], [21] for arithmetic operations. The reconfigurable multiple bus machine (RMBM) has been studied in [43], =-=[44]-=-, [45]. In this model, the reconfiguration hardware is separated from the processing elements. In [1], the Reconfigurable Network (RN) model is introduced and several algorithms for this model are der... |

19 |
Reconfigurable Buses With Shift Switching -- Concepts and Applications
- Lin, Olariu
- 1995
(Show Context)
Citation Context ...ic torus architecture [19] is identical to the PARBS except that the rows and columns of the underlying mesh wrap around (Fig. 5). A variant of the reconfigurable mesh model has been studied in [20], =-=[21]-=- for arithmetic operations. The reconfigurable multiple bus machine (RMBM) has been studied in [43], [44], [45]. In this model, the reconfiguration hardware is separated from the processing elements. ... |

15 |
A bit model of reconfigurable mesh
- Jang, Park, et al.
- 1994
(Show Context)
Citation Context ...LRN in [3]. In the MRN/LRN model, the number of possible connection patterns in each PE is 10. Fig. 6 shows the allowed connection patterns. A bit model of the reconfigurable mesh has been defined in =-=[14].s-=-JANG ET AL.: CONSTANT TIME ALGORITHMS FOR COMPUTATIONAL GEOMETRY ON THE RECONFIGURABLE MESH 3 Fig. 4. Connection patterns allowed in RMESH. Fig. 5. 4 – 4 polymorphic torus. Fig. 6. Connection patter... |

14 | Selection on the reconfigurable mesh
- Hao, MacKenzie, et al.
- 1992
(Show Context)
Citation Context ... parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection [7], =-=[9]-=-, multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry problems including co... |

14 |
The Image Understanding Architecture and its programming environment
- Weems, Burrill
- 1991
(Show Context)
Citation Context ...mented with broadcast buses [35], the bus automaton [39], the polymorphic-torus network [19], and the coterie network in the latest version of the Content Addressable Array Parallel Processor (CAAPP) =-=[50]-=-. A reconfigurable mesh has been built by NEC which has 512 PEs [18]. Researchers at IBM have implemented a reconfigurable mesh called polymorphictorus [19]. An optical implementation of the reconfigu... |

13 |
Determining the minimal-area encasing rectangle for an arbitrary closed curve
- Freeman, Shapira
- 1975
(Show Context)
Citation Context ...) time on an N – N reconfigurable mesh. Further, the extreme points can be stored in a row in angular order with respect to a point inside the convex hull. 3.2 Smallest Enclosing Box It is well know=-=n [8]-=- that the smallest enclosing rectangle of a set of N planar points has at least one side that is an extension of an edge of the convex hull. Hence, the smallest enclosing rectangle may be found by fir... |

13 | An optimal multiplication algorithm on reconfigurable mesh
- Jang, Park, et al.
- 1997
(Show Context)
Citation Context ... have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection [7], [9], multiplication =-=[13]-=-, division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry problems including convex hull, kdimension... |

12 |
A VLSI-optimal constant time Sorting on reconfigurable mesh
- Lin, Olariu, et al.
- 1992
(Show Context)
Citation Context ...etical side, many efficient parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], =-=[22]-=-, [46], selection [7], [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geom... |

12 |
Sorting n Numbers on n × n Reconfigurable Meshes with Buses
- Nigam, Sahni
- 1992
(Show Context)
Citation Context ... theoretical side, many efficient parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], =-=[29]-=-, [22], [46], selection [7], [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computationa... |

11 |
An o(log n) time parallel algorithm for triangulating a set of points in the plane. Information Processing Letters
- Wang, Tsin
- 1987
(Show Context)
Citation Context ...8. The dpolygon formed by two disjoint convex hulls having N1 and N2 extreme points can be triangulated in O(1) time using an N1 – N2 reconfigurable mesh. PROOF. We use the technique by Wang and Tsi=-=n [49]-=- on the CREW PRAM. Their technique results in an O(log N) time solution while the reconfigurable bus system can be exploited in nontrivial ways to achieve O(1) time. Without loss of generality, assume... |

10 |
Reconfigurable Mesh Algorithms for Image
- Jenq, Sahni
- 1991
(Show Context)
Citation Context ...reconfigurable mesh refers to the model defined in Section 2.1. 2.3 Some Basic Operations Several operations can be performed quickly on the reconfigurable mesh [1], [7], [9], [10], [12], [13], [15], =-=[16], [17-=-], [28], [32], [47]. We briefly outline the results used in our algorithms in Section 3. Given a 0/1 sequence, bj, 0 … j < N, the prefix-and computation is to compute, for each i, 0 … i < N, b0 »... |

8 |
A sub-logarithmic time sorting algorithm on a reconfigurable mesh
- Nakano, Masuzawa, et al.
- 1991
(Show Context)
Citation Context ...le mesh refers to the model defined in Section 2.1. 2.3 Some Basic Operations Several operations can be performed quickly on the reconfigurable mesh [1], [7], [9], [10], [12], [13], [15], [16], [17], =-=[28], [32],-=- [47]. We briefly outline the results used in our algorithms in Section 3. Given a 0/1 sequence, bj, 0 … j < N, the prefix-and computation is to compute, for each i, 0 … i < N, b0 » b1 » ž bi. ... |

8 |
Fast Computer Vision Algorithms for Reconfigurable Meshes
- Olariu, Schwing, et al.
- 1992
(Show Context)
Citation Context ...h. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection [7], [9], multiplication [13], division [34], histogramming [12], [17], =-=[32]-=-, and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry problems including convex hull, kdimensional maxima, two-set dominance counting, smallest... |

6 |
Efficient Parallel Algorithms for Some Geometric Problems on Reconfigurable Mesh
- Jang, Prasanna
- 1992
(Show Context)
Citation Context ... —————————— 1045-9219/97$10.00 ©1997 IEEE enclosing box, all-pairs nearest neighbor, and triangulation, all on a reconfigurable mesh of size N – N. Preliminary versions of this =-=paper have appeared in [11]-=-, [30], [31]. Previously, there has been a constant time algorithm for convex hull on a reconfigurable mesh [38]. The algorithm is flawed. We show a counter example and present a corrected and more ge... |

6 |
An Efficient Convex Hull Computation on the Reconfigurable Mesh
- Reisis
- 1992
(Show Context)
Citation Context ... reconfigurable mesh of size N – N. Preliminary versions of this paper have appeared in [11], [30], [31]. Previously, there has been a constant time algorithm for convex hull on a reconfigurable mes=-=h [38]-=-. The algorithm is flawed. We show a counter example and present a corrected and more general solution in Section 3.1. The minimal size of the reconfigurable mesh needed to solve the computational geo... |

5 |
Selection on the reconfigurable mesh
- ElGindy, Wegrowicz
- 1991
(Show Context)
Citation Context ...cient parallel algorithms have been developed for the reconfigurable mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection =-=[7]-=-, [9], multiplication [13], division [34], histogramming [12], [17], [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry problems includi... |

5 | Histogramming on a reconfigurable mesh computer
- Jenq, Sahni
- 1992
(Show Context)
Citation Context ...le mesh. These include algorithms for fundamental data movement operations [24], [27], sorting [10], [1], [29], [22], [46], selection [7], [9], multiplication [13], division [34], histogramming [12], =-=[17]-=-, [32], and graph problems [27], [47]. In this paper, we develop constant time algorithms for computational geometry problems including convex hull, kdimensional maxima, two-set dominance counting, sm... |

5 |
Asymptotically efficient hypercube algorithms for computational geometry
- MacKenzie, Stout
- 1990
(Show Context)
Citation Context ...h p ° S, find a q ° S, such that q › p and q is nearest to p among all points in S. Miller and Stout [25] solve the problem in O( N) time using log N levels of recursion on a 2-MCC of size N ¥ N =-=. In [23]-=-, Mackenzie and Stout solve it using log log N levels of recursion on a hypercube of size N to run in O(log N (log log N) 3 ) time. We develop an O(1) time algorithm for a reconfigurable mesh of size ... |

5 | Computational geometry on a reconfigurable mesh - Nigam, Sahni - 1994 |

4 | Constant-time convexity problems on reconfigurable meshes - Bokka, Gurla, et al. - 1995 |

4 |
Shift switching and novel arithmetic schemes
- Lin
- 1995
(Show Context)
Citation Context ...ymorphic torus architecture [19] is identical to the PARBS except that the rows and columns of the underlying mesh wrap around (Fig. 5). A variant of the reconfigurable mesh model has been studied in =-=[20]-=-, [21] for arithmetic operations. The reconfigurable multiple bus machine (RMBM) has been studied in [43], [44], [45]. In this model, the reconfiguration hardware is separated from the processing elem... |

4 |
List ranking and graph algorithms on the reconfigurable multiple bus machine
- Subbaraman, Trahan, et al.
- 1993
(Show Context)
Citation Context ...ying mesh wrap around (Fig. 5). A variant of the reconfigurable mesh model has been studied in [20], [21] for arithmetic operations. The reconfigurable multiple bus machine (RMBM) has been studied in =-=[43]-=-, [44], [45]. In this model, the reconfiguration hardware is separated from the processing elements. In [1], the Reconfigurable Network (RN) model is introduced and several algorithms for this model a... |

3 |
Optimal Simulations in Reconfigurable Arrays
- Ben-Asher, Gordon, et al.
- 1992
(Show Context)
Citation Context ...ements. In [1], the Reconfigurable Network (RN) model is introduced and several algorithms for this model are derived under the mesh restriction. This model has been denoted as MRN in [29] and LRN in =-=[3]-=-. In the MRN/LRN model, the number of possible connection patterns in each PE is 10. Fig. 6 shows the allowed connection patterns. A bit model of the reconfigurable mesh has been defined in [14].sJANG... |

3 | Configurational Computation: A New Computation Method on - Wang, Chen, et al. - 1991 |

2 |
A Reconfigurable Processor Array with Routing LSIs and General Purpose DSPs
- Levinson, Kuroda, et al.
- 1992
(Show Context)
Citation Context ...rphic-torus network [19], and the coterie network in the latest version of the Content Addressable Array Parallel Processor (CAAPP) [50]. A reconfigurable mesh has been built by NEC which has 512 PEs =-=[18]-=-. Researchers at IBM have implemented a reconfigurable mesh called polymorphictorus [19]. An optical implementation of the reconfigurable mesh is suggested by Ben-Asher and Schuster [2]. This implemen... |

2 |
Triangulation on a Reconfigurable Mesh with Buses
- Nigam, Sahni
- 1994
(Show Context)
Citation Context ...1045-9219/97$10.00 ©1997 IEEE enclosing box, all-pairs nearest neighbor, and triangulation, all on a reconfigurable mesh of size N – N. Preliminary versions of this paper have appeared in [11], [30=-=], [31]-=-. Previously, there has been a constant time algorithm for convex hull on a reconfigurable mesh [38]. The algorithm is flawed. We show a counter example and present a corrected and more general soluti... |

2 | Parallel Computations on Meshes with Static and Reconfigurable Buses - Reisis - 1989 |

1 | Optical Splitting Graphs - Ben-Asher, Schuster - 1990 |

1 | Approximate Coin Tossing with Applications to List, Tree and Graph problems - Cole, Vishkin - 1986 |

1 |
Reconfigurable Mesh Algorithms for the Area and
- Jenq, Sahni
- 1991
(Show Context)
Citation Context ... of the RMESH. It is assumed that processors on the same bus can communicate in O(1) time. RMESH algorithms for fundamental data movement operations and image processing problems can be found in [7], =-=[15], -=-[17], [24], [25], [27], [29], [38]. Fig. 3. 4 – 4 RMESH. The polymorphic torus architecture [19] is identical to the PARBS except that the rows and columns of the underlying mesh wrap around (Fig. 5... |

1 | A Content Addressable Array Parallel - Shu, Chow, et al. - 1988 |

1 |
Constant Time Graph and Poset Algorithms on the Reconfigurable Multiple Bus
- Trahan, Vaidyanathan, et al.
- 1994
(Show Context)
Citation Context ...ap around (Fig. 5). A variant of the reconfigurable mesh model has been studied in [20], [21] for arithmetic operations. The reconfigurable multiple bus machine (RMBM) has been studied in [43], [44], =-=[45]-=-. In this model, the reconfiguration hardware is separated from the processing elements. In [1], the Reconfigurable Network (RN) model is introduced and several algorithms for this model are derived u... |