## Mesh Connected Computers with Fixed and Reconfigurable Buses: Packet Routing, Sorting, and Selection (1993)

Venue: | In proc. 1st European Symp. on Algorithms |

Citations: | 25 - 10 self |

### BibTeX

@INPROCEEDINGS{Rajasekaran93meshconnected,

author = {Sanguthevar Rajasekaran},

title = {Mesh Connected Computers with Fixed and Reconfigurable Buses: Packet Routing, Sorting, and Selection},

booktitle = {In proc. 1st European Symp. on Algorithms},

year = {1993},

pages = {309--320}

}

### OpenURL

### Abstract

Mesh connected computers have become attractive models of computing because of their varied special features. In this paper we consider two variations of the mesh model: 1) a mesh with fixed buses, and 2) a mesh with reconfigurable buses. Both these models have been the subject matter of extensive previous research. We solve numerous important problems related to packet routing, sorting, and selection on these models. In particular, we provide lower bounds and very nearly matching upper bounds for the following problems on both these models: 1) Routing on a linear array; and 2) k \Gamma k routing and k \Gamma k sorting on a 2D mesh for any k 12. We provide an improved algorithm for 1 \Gamma 1 routing and a matching sorting algorithm. In addition we present greedy algorithms for 1 \Gamma 1 routing, k \Gamma k routing, and k \Gamma k sorting that are better on average and supply matching lower bounds. We also show that sorting can be performed in logarithmic time on a mesh with fixed bu...

### Citations

1308 |
Introduction to Parallel Algorithms and Architectures: Arrays
- Leighton
- 1992
(Show Context)
Citation Context ...g and sorting on the conventional mesh (see e.g., [56, 58, 50, 46, 47, 20, 21, 22, 25, 23, 42, 43, 41, 40, 38, 17, 16, 15]). An excellent reference for algorithms on the conventional mesh is Leighton =-=[24]-=-. Meshes with fixed buses have been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 18]). An equally impressive amount of work has been done on the mesh with reconfigu... |

713 |
A measure of asymptotic efficiency of tests of a hypothesis based on the sum of observations
- Chernoff
- 1952
(Show Context)
Citation Context ...head in a single flip being p. X is also known to have a binomial distribution B(n; p). The following three facts (known as Chernoff bounds) will be used in the paper (and were discovered by Chernoff =-=[5]-=- and Angluin & Valiant [1]): Prob:[Xsm]s` np m ' m e m\Gammanp ; Prob:[Xs(1 + ffl)np]sexp(\Gammaffl 2 np=3); and Prob:[Xs(1 \Gamma ffl)np]sexp(\Gammaffl 2 np=2); for any 0 ! ffl ! 1, and m ? np. Note:... |

369 | Time bounds for selection
- Blum, Floyd, et al.
- 1973
(Show Context)
Citation Context ...RBUS (from out of n elements) can be performed in O(1) time, provided we have an n \Theta n ffl mesh, ffl being any constant ? 0. Randomized selection algorithms have a rich history. Floyd and Rivest =-=[9]-=- presented an optimal randomized sequential algorithm for selection. Followed by this work, Reischuk [49] and Vishkin [59] showed how to perform selection in parallel on various models. Recently, Raja... |

256 |
Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings
- Angluin, Valiant
- 1979
(Show Context)
Citation Context ...g p. X is also known to have a binomial distribution B(n; p). The following three facts (known as Chernoff bounds) will be used in the paper (and were discovered by Chernoff [5] and Angluin & Valiant =-=[1]-=-): Prob:[Xsm]s` np m ' m e m\Gammanp ; Prob:[Xs(1 + ffl)np]sexp(\Gammaffl 2 np=3); and Prob:[Xs(1 \Gamma ffl)np]sexp(\Gammaffl 2 np=2); for any 0 ! ffl ! 1, and m ? np. Note: Most of our 2D routing al... |

85 | The Power of Reconfiguration
- Ben-Asher
- 1991
(Show Context)
Citation Context ...n previous works as well (see e.g., [11, 10]). Both M r and M f are becoming popular models of computing because of the absence of diameter consideration and because of the commercial implementations =-=[2, 28, 55, 27]-=-. Even as theoretical models, both M r and M f are very interesting. For instance, n keys can be sorted in O(1) time on an n \Theta n mesh M r , whereas we know that\Omega\Gamma log n log log n ) time... |

60 |
Average Case Analysis of Greedy Routing algorithms on Arrays
- Leighton
- 1990
(Show Context)
Citation Context ...is needed even on the CRCW PRAM given only a polynomial number of processors. 1.5 Previous and New Results Numerous papers have been written on routing and sorting on the conventional mesh (see e.g., =-=[56, 58, 50, 46, 47, 20, 21, 22, 25, 23, 42, 43, 41, 40, 38, 17, 16, 15]-=-). An excellent reference for algorithms on the conventional mesh is Leighton [24]. Meshes with fixed buses have been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 1... |

53 |
Meshes with reconfigurable buses
- Miller
- 1988
(Show Context)
Citation Context ...been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 18]). An equally impressive amount of work has been done on the mesh with reconfigurable buses as well (see e.g., =-=[2, 29, 30, 51, 60, 31, 33, 34]-=-). Mesh with fixed buses Leung and Shende [26] have shown that on a linear array with a bus, permutation routing needs 2n 3 steps in the worst case and presented a 2n 3 step algorithm as well. They al... |

42 |
Array Processor with Multiple Broadcasting
- Kumar, Raghavendra
- 1987
(Show Context)
Citation Context ... 43, 41, 40, 38, 17, 16, 15]). An excellent reference for algorithms on the conventional mesh is Leighton [24]. Meshes with fixed buses have been studied by various researchers in the past (see e.g., =-=[52, 53, 3, 19, 54, 4, 26, 18]-=-). An equally impressive amount of work has been done on the mesh with reconfigurable buses as well (see e.g., [2, 29, 30, 51, 60, 31, 33, 34]). Mesh with fixed buses Leung and Shende [26] have shown ... |

40 |
Finding Maximum on an Array Processor with a Global Bus
- Bokhari
- 1984
(Show Context)
Citation Context ... 43, 41, 40, 38, 17, 16, 15]). An excellent reference for algorithms on the conventional mesh is Leighton [24]. Meshes with fixed buses have been studied by various researchers in the past (see e.g., =-=[52, 53, 3, 19, 54, 4, 26, 18]-=-). An equally impressive amount of work has been done on the mesh with reconfigurable buses as well (see e.g., [2, 29, 30, 51, 60, 31, 33, 34]). Mesh with fixed buses Leung and Shende [26] have shown ... |

40 |
Concentrated Regular Data Streams on Grids: Sorting and Routing Near to the Bisection Bound
- Kunde
- 1991
(Show Context)
Citation Context ...he blocks are ordered in snake-like row-major order. The ordering of keys within a block is immaterial. Keys in any block will besall the keys in the next higher block. A similar indexing was used in =-=[22]-=- also. Figure 1 gives an example of a 6 \Theta 6 mesh with 2 keys per node. Blocks are of size 2 \Theta 2. Given a set of N numbers and an isN , the problem of selection is to find the ith smallest el... |

36 |
A Fast Algorithm for Computing Histograms on a Reconfigurable Mesh
- Jang, Park, et al.
- 1992
(Show Context)
Citation Context ... applications. Similar, but not exactly the same, Lemmas have been proven in [51]. 9.1 Some Basics The following Lemmas will be employed in our selection algorithm: Lemma 9.2 Jang, Park, and Prasanna =-=[11]-=-: If each node in an n \Theta n mesh M r has a bit, the number of 1's can be computed in O(log n) time. Lemma 9.3 Jang and Prasanna [12]: For any 1srsn, elements in the first r rows of an n \Theta n m... |

35 |
Matching the bisection bound for routing and sorting on the mesh
- Kaufmann, Rajasekaran, et al.
- 1992
(Show Context)
Citation Context ...is needed even on the CRCW PRAM given only a polynomial number of processors. 1.5 Previous and New Results Numerous papers have been written on routing and sorting on the conventional mesh (see e.g., =-=[56, 58, 50, 46, 47, 20, 21, 22, 25, 23, 42, 43, 41, 40, 38, 17, 16, 15]-=-). An excellent reference for algorithms on the conventional mesh is Leighton [24]. Meshes with fixed buses have been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 1... |

32 |
A 2n \Gamma 2 Step Algorithm for Routing in an n \Theta n Array with Constant Size Queues
- Leighton, Makedon, et al.
- 1989
(Show Context)
Citation Context ...is needed even on the CRCW PRAM given only a polynomial number of processors. 1.5 Previous and New Results Numerous papers have been written on routing and sorting on the conventional mesh (see e.g., =-=[56, 58, 50, 46, 47, 20, 21, 22, 25, 23, 42, 43, 41, 40, 38, 17, 16, 15]-=-). An excellent reference for algorithms on the conventional mesh is Leighton [24]. Meshes with fixed buses have been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 1... |

24 |
Sorting n numbers on n \Theta n reconfigurable meshes with buses
- Nigam, Sahni
- 1994
(Show Context)
Citation Context ...Reconfigurable Mesh A number of interesting results have been obtained on the reconfigurable mesh. Some of them are: 1) Prefix and other problems [29]; 2) Constant time sorting and routing algorithms =-=[32, 2, 31, 60]-=-; 3) Random Access Read and Random Access Write [29]; 4) Histogramming and Related Problems [13, 14, 11]; 5) Permutation Routing and Sorting [41, 18]; 6) Selection [8, 10, 7]. In this paper we show th... |

22 |
Randomized Parallel Selection
- Rajasekaran
- 1990
(Show Context)
Citation Context ...ed an optimal randomized sequential algorithm for selection. Followed by this work, Reischuk [49] and Vishkin [59] showed how to perform selection in parallel on various models. Recently, Rajasekaran =-=[37]-=- gave an optimal selection algorithm for the hypercube. More recently, Kaklamanis, et. al. [16] have presented an efficient selection algorithm for the MIMD mesh that runs in 1:22n + e o(n) steps. All... |

21 |
Optimal Sorting on Mesh-Connected Processor Arrays
- Kaklamanis, Krizanc
- 1992
(Show Context)
Citation Context |

19 | Routing and Sorting on Mesh Connected Processor Arrays - Kunde - 1988 |

16 |
Efficient median finding and its application to two-variable linear programming on mesh connected computers with multiple broadcasting
- Chen, Chen
- 1992
(Show Context)
Citation Context ... 43, 41, 40, 38, 17, 16, 15]). An excellent reference for algorithms on the conventional mesh is Leighton [24]. Meshes with fixed buses have been studied by various researchers in the past (see e.g., =-=[52, 53, 3, 19, 54, 4, 26, 18]-=-). An equally impressive amount of work has been done on the mesh with reconfigurable buses as well (see e.g., [2, 29, 30, 51, 60, 31, 33, 34]). Mesh with fixed buses Leung and Shende [26] have shown ... |

15 |
Fundamental algorithms on reconfigurable meshes
- Olariu, Schwing, et al.
- 1991
(Show Context)
Citation Context ...been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 18]). An equally impressive amount of work has been done on the mesh with reconfigurable buses as well (see e.g., =-=[2, 29, 30, 51, 60, 31, 33, 34]-=-). Mesh with fixed buses Leung and Shende [26] have shown that on a linear array with a bus, permutation routing needs 2n 3 steps in the worst case and presented a 2n 3 step algorithm as well. They al... |

11 |
Comprehensive Evaluation of a Two-Dimensional Configurable Array
- Menzilcioglu, Kung, et al.
- 1989
(Show Context)
Citation Context ...n previous works as well (see e.g., [11, 10]). Both M r and M f are becoming popular models of computing because of the absence of diameter consideration and because of the commercial implementations =-=[2, 28, 55, 27]-=-. Even as theoretical models, both M r and M f are very interesting. For instance, n keys can be sorted in O(1) time on an n \Theta n mesh M r , whereas we know that\Omega\Gamma log n log log n ) time... |

10 |
Reconfigurable Mesh Algorithms for Image
- Jenq, Sahni
- 1991
(Show Context)
Citation Context ...them are: 1) Prefix and other problems [29]; 2) Constant time sorting and routing algorithms [32, 2, 31, 60]; 3) Random Access Read and Random Access Write [29]; 4) Histogramming and Related Problems =-=[13, 14, 11]-=-; 5) Permutation Routing and Sorting [41, 18]; 6) Selection [8, 10, 7]. In this paper we show that k \Gamma k routing and k \Gamma k sorting need kn 2 steps on M r and show that there exist algorithms... |

10 |
Multi-Packet Routing on Mesh Connected Arrays
- Kunde, Tensi
- 1989
(Show Context)
Citation Context |

10 |
Packet routing on square meshes with row and column buses
- Leung, Shende
- 1991
(Show Context)
Citation Context |

9 | Permutation Routing on Reconfigurable Meshes - Cogolludo, Rajasekaran - 2001 |

8 |
The light bulb problem
- PATURI, RAJASEKARAN, et al.
- 1995
(Show Context)
Citation Context ...show that on average it only takes (3\Gamma p 5)n 2 �� :382n steps. We also prove similar results for routing on a 2D mesh, k \Gamma k routing, and k \Gamma k sorting. The following Lemmas (see e.=-=g., [35, 24]-=-) will prove helpful in our analysis: Let z 1 ; z 2 ; : : : ; z m be 0; 1 valued independent random variables such that Prob:[z j = 1] = p j for 1sjsm. Let S m = P m j=1 z j and the expectation of S m... |

7 |
An Optimal Sorting Algorithm on Reconfigurable
- Jang, Prasanna
- 1992
(Show Context)
Citation Context ... in our selection algorithm: Lemma 9.2 Jang, Park, and Prasanna [11]: If each node in an n \Theta n mesh M r has a bit, the number of 1's can be computed in O(log n) time. Lemma 9.3 Jang and Prasanna =-=[12]-=-: For any 1srsn, elements in the first r rows of an n \Theta n mesh M r can be sorted in O(r) time. Problem 4. Consider an n \Theta n mesh M r . Say there are ` i elements arbitrarily distributed in r... |

6 | Permutations on illiac iv-type networks - Raghavendra, Kumar - 1986 |

5 |
Applications of reconfigurable meshes to constant-time computations
- Olariu, Schwing, et al.
- 1993
(Show Context)
Citation Context ...been studied by various researchers in the past (see e.g., [52, 53, 3, 19, 54, 4, 26, 18]). An equally impressive amount of work has been done on the mesh with reconfigurable buses as well (see e.g., =-=[2, 29, 30, 51, 60, 31, 33, 34]-=-). Mesh with fixed buses Leung and Shende [26] have shown that on a linear array with a bus, permutation routing needs 2n 3 steps in the worst case and presented a 2n 3 step algorithm as well. They al... |

4 |
Three Algorithms for Selection on the Reconfigurable Mesh
- Doctor, Krizanc
- 1993
(Show Context)
Citation Context ...d routing algorithms [32, 2, 31, 60]; 3) Random Access Read and Random Access Write [29]; 4) Histogramming and Related Problems [13, 14, 11]; 5) Permutation Routing and Sorting [41, 18]; 6) Selection =-=[8, 10, 7]-=-. In this paper we show that k \Gamma k routing and k \Gamma k sorting need kn 2 steps on M r and show that there exist algorithms for these problems with very nearly matching time bounds, for any ks8... |

4 |
Constant-time Sorting on a Reconfigurable Mesh
- Nakano, Peleg, et al.
- 1992
(Show Context)
Citation Context |

2 |
A comparison of meshes with static buses and unidirectional wrap-arounds, Parallel Processing
- Krizanc, Rajasekaran, et al.
- 1993
(Show Context)
Citation Context ...routing on a reconfigurable mesh, where it was shown that sorting can be reduced to routing easily if there exists a mechanism for broadcasting. Using this reduction, Krizanc, Rajasekaran, and Shende =-=[18]-=- have given an algorithm for M f that runs in time n+4 n q +e o(n), the queue size being 2q. In our algorithm also we adopt this reduction. We only provide a summary of the algorithm, due to space con... |

1 |
Histogramming on a Reconfigurable Mesh
- Jenq, Sahni
- 1992
(Show Context)
Citation Context ...them are: 1) Prefix and other problems [29]; 2) Constant time sorting and routing algorithms [32, 2, 31, 60]; 3) Random Access Read and Random Access Write [29]; 4) Histogramming and Related Problems =-=[13, 14, 11]-=-; 5) Permutation Routing and Sorting [41, 18]; 6) Selection [8, 10, 7]. In this paper we show that k \Gamma k routing and k \Gamma k sorting need kn 2 steps on M r and show that there exist algorithms... |