### Abstract

Sorting is an important subroutine in many parallel algorithms and has been studied extensively on meshes and related networks. If every processor of an n × n mesh is the source and destination of at most k elements, then sorting requires at least k k; n/2 steps in the worst-case, and simple algorithms have recently been proposed that nearly match this bound. However, this lower bound does not extend to non-worst-case inputs, or weaker definitions of sorting that are sufficient in many applications. In this paper, we give algorithms and lower bounds for several such problems. We first present a very simple scheme for k-k routing that performs optimally under both average-case and worst-case inputs. As an application of this scheme, we describe a simple k-k sorting algorithm based on sample sort that nearly matches this bound. The main part of the paper considers several 'sorting-like' problems. In the ranking problem, the ranks of all elements have to be determined, but there is no requirement about their final positions. We describe an algorithm running in time (1 +o(l)) k n/4 steps, which is nearly optimal under the considered model of the mesh. We show that integer versions of the sorting and ranking problems, where keys are drawn from {0,…, m — 1}, can be solved asymptotically faster than the general problems for small values of m. A related problem, the excess counting problem, can be solved in O(n) steps in many interesting cases.

Original language | English (US) |
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Title of host publication | Algorithms - ESA 1995 - 3rd Annual European Symposium, Proceedings |

Publisher | Springer Verlag |

Pages | 75-88 |

Number of pages | 14 |

Volume | 979 |

ISBN (Print) | 3540603131, 9783540603139 |

State | Published - 1995 |

Event | 3rd Annual European Symposium on Algorithms, ESA 1995 - Corfu, Greece Duration: Sep 25 1995 → Sep 27 1995 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 979 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 3rd Annual European Symposium on Algorithms, ESA 1995 |
---|---|

Country | Greece |

City | Corfu |

Period | 9/25/95 → 9/27/95 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Algorithms - ESA 1995 - 3rd Annual European Symposium, Proceedings*(Vol. 979, pp. 75-88). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 979). Springer Verlag.

**Beyond the worst-case bisection bound : Fast sorting and ranking on meshes.** / Kaufmann, Michael; Sibeyn, Jop F.; Suel, Torsten.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms - ESA 1995 - 3rd Annual European Symposium, Proceedings.*vol. 979, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 979, Springer Verlag, pp. 75-88, 3rd Annual European Symposium on Algorithms, ESA 1995, Corfu, Greece, 9/25/95.

}

TY - GEN

T1 - Beyond the worst-case bisection bound

T2 - Fast sorting and ranking on meshes

AU - Kaufmann, Michael

AU - Sibeyn, Jop F.

AU - Suel, Torsten

PY - 1995

Y1 - 1995

N2 - Sorting is an important subroutine in many parallel algorithms and has been studied extensively on meshes and related networks. If every processor of an n × n mesh is the source and destination of at most k elements, then sorting requires at least k k; n/2 steps in the worst-case, and simple algorithms have recently been proposed that nearly match this bound. However, this lower bound does not extend to non-worst-case inputs, or weaker definitions of sorting that are sufficient in many applications. In this paper, we give algorithms and lower bounds for several such problems. We first present a very simple scheme for k-k routing that performs optimally under both average-case and worst-case inputs. As an application of this scheme, we describe a simple k-k sorting algorithm based on sample sort that nearly matches this bound. The main part of the paper considers several 'sorting-like' problems. In the ranking problem, the ranks of all elements have to be determined, but there is no requirement about their final positions. We describe an algorithm running in time (1 +o(l)) k n/4 steps, which is nearly optimal under the considered model of the mesh. We show that integer versions of the sorting and ranking problems, where keys are drawn from {0,…, m — 1}, can be solved asymptotically faster than the general problems for small values of m. A related problem, the excess counting problem, can be solved in O(n) steps in many interesting cases.

AB - Sorting is an important subroutine in many parallel algorithms and has been studied extensively on meshes and related networks. If every processor of an n × n mesh is the source and destination of at most k elements, then sorting requires at least k k; n/2 steps in the worst-case, and simple algorithms have recently been proposed that nearly match this bound. However, this lower bound does not extend to non-worst-case inputs, or weaker definitions of sorting that are sufficient in many applications. In this paper, we give algorithms and lower bounds for several such problems. We first present a very simple scheme for k-k routing that performs optimally under both average-case and worst-case inputs. As an application of this scheme, we describe a simple k-k sorting algorithm based on sample sort that nearly matches this bound. The main part of the paper considers several 'sorting-like' problems. In the ranking problem, the ranks of all elements have to be determined, but there is no requirement about their final positions. We describe an algorithm running in time (1 +o(l)) k n/4 steps, which is nearly optimal under the considered model of the mesh. We show that integer versions of the sorting and ranking problems, where keys are drawn from {0,…, m — 1}, can be solved asymptotically faster than the general problems for small values of m. A related problem, the excess counting problem, can be solved in O(n) steps in many interesting cases.

UR - http://www.scopus.com/inward/record.url?scp=84947737984&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947737984&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84947737984

SN - 3540603131

SN - 9783540603139

VL - 979

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 75

EP - 88

BT - Algorithms - ESA 1995 - 3rd Annual European Symposium, Proceedings

PB - Springer Verlag

ER -