## Deterministic Permutation Routing on Meshes (1993)

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Venue: | Proc. 5th Symp. on Parallel and Distributed Proc., IEEE |

Citations: | 20 - 13 self |

### BibTeX

@INPROCEEDINGS{Chlebus93deterministicpermutation,

author = {Bogdan Chlebus and Michael Kaufmann and Jop F. Sibeyn},

title = {Deterministic Permutation Routing on Meshes},

booktitle = {Proc. 5th Symp. on Parallel and Distributed Proc., IEEE},

year = {1993},

pages = {814--821}

}

### OpenURL

### Abstract

We present a new deterministic algorithm for routing permutations on a two-dimensional MIMD mesh. The algorithm runs in the optimal time 2n \Gamma 2 on an n \Theta n mesh, and the maximal number of packets stored in a processing unit is 81. A modification of the algorithm, running in time 2n + O(1), has the maximal queue length of only 31. The algorithm is simple, no conflict-resolution strategy is required. Keywords: mesh-connected computer, permutation routing, optimal algorithm, conflict freeness. 1 Introduction The exchange of information between processing units, PUs, is one of the most important problems on parallel computers in which the PUs communicate through an interconnection network. The basic communication step is that of transferring packets. These are portions of information generated and received by This research was partially supported by EC Cooperative Action IC-1000 (project ALTEC: Algorithms for Future Technologies). y Instytut Informatyki, Uniwersytet Warszaws...

### Citations

1374 |
Introduction to Parallel Algorithms and Architectures: Arrays
- LEIGHTON
- 1992
(Show Context)
Citation Context ... routing algorithm is called optimal if T = 2 \Delta n \Gamma 2, and near-optimal if T = 2 \Delta n +O(1). Recently, routing on two-dimensional meshes has attracted a considerable amount of attention =-=[6]-=-. The first routing algorithms which required close to 2 \Delta n steps were given by Kunde [4] and Rajasekaran and Tsantilas [8]. Leighton, Makedon and Tollis [5] presented the first deterministic al... |

93 |
and H.T.Kung, "Sorting on a mesh-connected parallel computer
- Thompson
- 1977
(Show Context)
Citation Context ...4 (5 \Delta 48 = 240 ? 224). This means that they still have to apply recursion one phase earlier, i.e. for some ss32. In [9] the scattering is performed by the sorting algorithm of Thompson and Kung =-=[14]-=- for which the time consumption can be estimated by T ss3 \Delta s + (3 + 3=2 \Delta log s) \Delta s 2=3 . However, this is for sorting in a snake-like order. For the estimate of the maximal queue siz... |

90 |
An optical sorting algorithm for meshconnected computers
- Schnorr, Shamir
- 1986
(Show Context)
Citation Context ...ring for Near-Optimal Routing In this section scattering algorithm scatter1 is developed, which is used with the near-optimal algorithms. It is inspired by the sorting algorithm of Schnorr and Shamir =-=[9]-=-. It takes only 2 \Delta s +O(s 7=10 ) steps. The scattering is as good as obtained by sorting s \Theta s squares in column-major order. A similar method was used by Gu and Gu [2]. Each s \Theta s squ... |

41 |
Concentrated regular data streams on grids: Sorting and routing near to the bisection bound
- Kunde
- 1991
(Show Context)
Citation Context ...orly sorting the packets in column-major order was used as a scattering technique with a comparable efficiency, and it could be performed in time 3 1 = 2 \Delta s+o(s) applying the algorithm of Kunde =-=[5]-=-. For the optimal time algorithm, we scatter in 6 \Delta s steps for all necessary s. In [9], the bound on the scattering time (sorting in their case) is not handled very carefully. With a scarce expl... |

36 |
Matching the bisection bound for routing and sorting on the mesh
- Kaufmann, Rajasekaran, et al.
- 1992
(Show Context)
Citation Context ...he task of sorting and routing, when both the initial and final distributions of packets are to be k-distributions, is called k-k sorting and k-k routing, respectively. The k-k routing was studied in =-=[1, 3]-=-. Consider a linear array of n PUs numbered P 0 through Pn\Gamma1 . We will develop optimal sorting for this model. First we give a lower bound. For a given input k-distribution D definesg(i; j), for ... |

36 |
Optimal routing algorithms for meshconnected processor arrays
- Rajasekaran, Tsantilas
- 1992
(Show Context)
Citation Context ...wo-dimensional meshes has attracted a considerable amount of attention [6]. The first routing algorithms which required close to 2 \Delta n steps were given by Kunde [4] and Rajasekaran and Tsantilas =-=[8]-=-. Leighton, Makedon and Tollis [5] presented the first deterministic algorithm with optimal routing time and constant size queues. This paper was of a great theoretical importance but the maximum queu... |

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 ...d a considerable amount of attention [6]. The first routing algorithms which required close to 2 \Delta n steps were given by Kunde [4] and Rajasekaran and Tsantilas [8]. Leighton, Makedon and Tollis =-=[5]-=- presented the first deterministic algorithm with optimal routing time and constant size queues. This paper was of a great theoretical importance but the maximum queue size was impractically large (Q ... |

32 | Constant Queue Routing on a Mesh
- Rajasekaran, Overholt
- 1990
(Show Context)
Citation Context ... deterministic algorithm with optimal routing time and constant size queues. This paper was of a great theoretical importance but the maximum queue size was impractically large (Q = 1008 according to =-=[7]-=-). Rajasekaran and Overholt [7] reduced Q, though the claimed Q = 112 was not correct (see [1]). Gu and Gu [2] developed a similar algorithm, claiming it to have queue size Q = 96 for the optimal vers... |

20 |
Routing and Sorting on Mesh Connected Processor Arrays
- Kunde
(Show Context)
Citation Context ...\Delta n \Gamma 2 is optimal. In the past years, routing and sorting on meshes have attracted a considerable amount of attention [6]. The first near-optimal algorithms for routing were given by Kunde =-=[2]-=- and Rajasekaran and Tsantilas [8]. Later Leighton, Makedon and Tollis [5] presented 1 the first deterministic algorithm with optimal routing time and constant size queues. This paper is of great theo... |

18 | Supporting the hypercube programming model on mesh architectures
- Stricker
- 1992
(Show Context)
Citation Context ...n this algorithm has to be uni-axial, that is, it cannot use horizontal and vertical links simultaneously. None of the known sorting algorithms appears to perform well for small size meshes. Stricker =-=[13]-=- points out the algorithm RevSort of [11] as an exception, but this algorithm requires more than 2 1 = 2 \Delta log log s+3 \Delta s steps for sorting in snake-like order, and is not uni-axial. Our al... |

15 | Desnakification of Mesh Sorting Algorithms
- Sibeyn
- 1994
(Show Context)
Citation Context ...of scattering algorithms that give competitive queue sizes. For near-optimal algorithms, a scattering technique is designed, with T ss2 \Delta s+o(s). For optimal algorithms, a sorting algorithm from =-=[10]-=- is used. In the basic algorithms of this paper, developed in Sections 4, 5, and 6, packets influence each other only during the scattering phase: they are routed locally depending on their destinatio... |

11 |
Systolic sorting on a mesh connected networks
- Lang, Schimmler, et al.
- 1985
(Show Context)
Citation Context ...orithm requires more than 2 1 = 2 \Delta log log s+3 \Delta s steps for sorting in snake-like order, and is not uni-axial. Our algorithms borrow some ideas from the recursive algorithm of Lang et al. =-=[6]-=-. The following results are also interesting for their own sake; for instance, with a factor 2 delay the algorithm can be used for sorting in row-major order on an SIMD mesh. The algorithms sort2 and ... |

9 | Towards practical permutation routing on meshes
- Kaufmann, Meyer, et al.
- 1994
(Show Context)
Citation Context ...teps with Q = 32, another has much smaller queues, Q = 12, but T = 2 \Delta n +O(1). The first algorithm may be practical. Actually, we successfully implemented it on our mesh simulator (described in =-=[3]-=-). On the other hand, the constant hidden in the O(1) is prohibitive, approximately 10 7 . Therefore, the second result has merely theoretical importance: the algorithm makes a major step towards near... |

7 | Shorter Queues for Permutation Routing on Meshes
- Sibeyn, Chlebus, et al.
- 1994
(Show Context)
Citation Context ...he spreading techniques and the most queue-size efficient algorithms of this paper. The conclusions and points for further research are given in Section 8. The results of this paper were announced in =-=[1, 11]-=- in a preliminary form. 2 Preliminaries A two-dimensional n \Theta n processor array without wrap-around connections is assumed as the computer model. This machine is referred to simply as mesh. It co... |

2 |
Two Packet Routing Algorithms on a Mesh-Connected Computer
- Gu, Gu
- 1993
(Show Context)
Citation Context ...ical importance but the maximum queue size was impractically large (Q = 1008 according to [7]). Rajasekaran and Overholt [7] reduced Q, though the claimed Q = 112 was not correct (see [1]). Gu and Gu =-=[2]-=- developed a similar algorithm, claiming it to have queue size Q = 96 for the optimal version, and Q = 26 for the near-optimal case. Their paper, however, also contained flaws. There seem to be many p... |

1 |
Routing and Sorting on Mesh Connected
- Kunde
- 1988
(Show Context)
Citation Context ...ta n +O(1). Recently, routing on two-dimensional meshes has attracted a considerable amount of attention [6]. The first routing algorithms which required close to 2 \Delta n steps were given by Kunde =-=[4]-=- and Rajasekaran and Tsantilas [8]. Leighton, Makedon and Tollis [5] presented the first deterministic algorithm with optimal routing time and constant size queues. This paper was of a great theoretic... |

1 |
Note on the Paper of Gu and Gu', manuscript
- Sibeyn, Kaufmann
- 1993
(Show Context)
Citation Context ...ptimal version. Their paper contains serious flaws however. A detailed exposition of an example of a packet routing problem for which the algorithm exceeds the claimed resource bounds can be found in =-=[12]-=-. We give new routing algorithms with smaller bounds on the queue size. One of them operates in time T = 2 \Delta n \Gamma 2 with Q = 81, another has much smaller queues, Q = 31, but T = 2 \Delta n+O(... |