### Abstract

This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We show that in any network in which each physical channel can emulate up to Q virtual channels, it is possible to route any set of L-bit messages whose paths have congestion C and dilation D in (L+D)C(D log D)^{1/Q}2^{O(log*(C/D))} bit steps. We also prove a nearly matching lower bound, i.e., for any values of C, D, Q, and L, where C, D≥Q+1 and L = (1+Ω(1))D, we show how to construct a network and a set of L-bit messages whose paths have congestion C and dilation D that require Ω(LCD^{1/Q}) bit steps to route. These upper and lower bounds imply that increasing the queuing capacity Q of each physical channel can speed up a wormhole routing algorithm by a superlinear factor. The results can be translated to the scenario in which each physical channel can transmit B bits simultaneously, and can queue bits from B different messages. In this case, the bounds are (L+D)C(D log D)^{1/B}2^{O(log* (C/D))}/B and Ω(LCD^{1/B}/B), respectively. We also present a simple randomized wormhole routing algorithm for the butterfly network. The algorithm routes a q-relation on the inputs and outputs of an n-input butterfly in O(LQ(q+log n)(log^{1/Q} n) log log(qn)) bit-steps. We present a nearly-matching lower bound that holds for a broad class of algorithms.

Original language | English (US) |
---|---|

Title of host publication | Annual ACM Symposium on Parallel Algorithms and Architectures |

Editors | Anon |

Pages | 131-141 |

Number of pages | 11 |

State | Published - 1996 |

Event | Proceedings of the 1996 8th Annual ACM Symposium on Parallel Algorithms and Architectures - Padua, Italy Duration: Jun 24 1996 → Jun 26 1996 |

### Other

Other | Proceedings of the 1996 8th Annual ACM Symposium on Parallel Algorithms and Architectures |
---|---|

City | Padua, Italy |

Period | 6/24/96 → 6/26/96 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Safety, Risk, Reliability and Quality

### Cite this

*Annual ACM Symposium on Parallel Algorithms and Architectures*(pp. 131-141)

**On the benefit of supporting virtual channels in wormhole routers.** / Cole, Richard; Maggs, Bruce M.; Sitaraman, Ramesh K.

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

*Annual ACM Symposium on Parallel Algorithms and Architectures.*pp. 131-141, Proceedings of the 1996 8th Annual ACM Symposium on Parallel Algorithms and Architectures, Padua, Italy, 6/24/96.

}

TY - GEN

T1 - On the benefit of supporting virtual channels in wormhole routers

AU - Cole, Richard

AU - Maggs, Bruce M.

AU - Sitaraman, Ramesh K.

PY - 1996

Y1 - 1996

N2 - This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We show that in any network in which each physical channel can emulate up to Q virtual channels, it is possible to route any set of L-bit messages whose paths have congestion C and dilation D in (L+D)C(D log D)1/Q2O(log*(C/D)) bit steps. We also prove a nearly matching lower bound, i.e., for any values of C, D, Q, and L, where C, D≥Q+1 and L = (1+Ω(1))D, we show how to construct a network and a set of L-bit messages whose paths have congestion C and dilation D that require Ω(LCD1/Q) bit steps to route. These upper and lower bounds imply that increasing the queuing capacity Q of each physical channel can speed up a wormhole routing algorithm by a superlinear factor. The results can be translated to the scenario in which each physical channel can transmit B bits simultaneously, and can queue bits from B different messages. In this case, the bounds are (L+D)C(D log D)1/B2O(log* (C/D))/B and Ω(LCD1/B/B), respectively. We also present a simple randomized wormhole routing algorithm for the butterfly network. The algorithm routes a q-relation on the inputs and outputs of an n-input butterfly in O(LQ(q+log n)(log1/Q n) log log(qn)) bit-steps. We present a nearly-matching lower bound that holds for a broad class of algorithms.

AB - This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We show that in any network in which each physical channel can emulate up to Q virtual channels, it is possible to route any set of L-bit messages whose paths have congestion C and dilation D in (L+D)C(D log D)1/Q2O(log*(C/D)) bit steps. We also prove a nearly matching lower bound, i.e., for any values of C, D, Q, and L, where C, D≥Q+1 and L = (1+Ω(1))D, we show how to construct a network and a set of L-bit messages whose paths have congestion C and dilation D that require Ω(LCD1/Q) bit steps to route. These upper and lower bounds imply that increasing the queuing capacity Q of each physical channel can speed up a wormhole routing algorithm by a superlinear factor. The results can be translated to the scenario in which each physical channel can transmit B bits simultaneously, and can queue bits from B different messages. In this case, the bounds are (L+D)C(D log D)1/B2O(log* (C/D))/B and Ω(LCD1/B/B), respectively. We also present a simple randomized wormhole routing algorithm for the butterfly network. The algorithm routes a q-relation on the inputs and outputs of an n-input butterfly in O(LQ(q+log n)(log1/Q n) log log(qn)) bit-steps. We present a nearly-matching lower bound that holds for a broad class of algorithms.

UR - http://www.scopus.com/inward/record.url?scp=0030387157&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030387157&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0030387157

SP - 131

EP - 141

BT - Annual ACM Symposium on Parallel Algorithms and Architectures

A2 - Anon, null

ER -