## Abstract

A good measure of the rate of convergence of path-vector protocols is the number of synchronous iterations required for convergence in the worst case. From an algebraic perspective, the rate of convergence depends on the expressive power of the routing algebra associated with the protocol. For example in a network of n nodes, shortest-path protocols are guaranteed to converge in O(n) iterations. In contrast the algebra underlying the Border Gateway Protocol (BGP) is in some sense too expressive and the protocol is not guaranteed to converge.There is significant interest in finding well-behaved algebras that still have enough expressive power to satisfy network operators.

Recent theoretical results have shown that by constraining routing algebras to those that are ``strictly increasing'' we can guarantee the convergence of path-vector protocols. Currently the best theoretical worst-case upper bound for the convergence of such algebras is $O(n!)$ iterations. However in practice it is difficult to find examples that do not converge in $n$ iterations. In this paper we close this gap. We first present a family of network configurations that converges in $\Theta(n^2)$ iterations, demonstrating that the worst case is $\Omega(n^2)$ iterations. We then prove that path-vector protocols

with a strictly increasing algebra are guaranteed to converge in $O(n^2)$ iterations. Together these results establish a tight $\Theta(n^2)$ bound. This is another piece of the puzzle in showing that ``strictly increasing" is, at least on a technical level, a reasonable constraint for practical policy-rich protocols.

Recent theoretical results have shown that by constraining routing algebras to those that are ``strictly increasing'' we can guarantee the convergence of path-vector protocols. Currently the best theoretical worst-case upper bound for the convergence of such algebras is $O(n!)$ iterations. However in practice it is difficult to find examples that do not converge in $n$ iterations. In this paper we close this gap. We first present a family of network configurations that converges in $\Theta(n^2)$ iterations, demonstrating that the worst case is $\Omega(n^2)$ iterations. We then prove that path-vector protocols

with a strictly increasing algebra are guaranteed to converge in $O(n^2)$ iterations. Together these results establish a tight $\Theta(n^2)$ bound. This is another piece of the puzzle in showing that ``strictly increasing" is, at least on a technical level, a reasonable constraint for practical policy-rich protocols.

Original language | English |
---|---|

Title of host publication | Proceedings - International Conference on Network Protocols, ICNP |

Place of Publication | USA |

Publisher | IEEE, Institute of Electrical and Electronics Engineers |

Pages | 335-345 |

Number of pages | 11 |

ISBN (Electronic) | 9781538660430 |

ISBN (Print) | 978-153866043-0 |

DOIs | |

Publication status | Published - 7 Nov 2018 |

Externally published | Yes |

Event | 26th International Conference on Network Protocols 2018 - Cambridge, United Kingdom Duration: 25 Sept 2018 → 27 Sept 2018 |

### Publication series

Name | Proceedings - International Conference on Network Protocols, ICNP |
---|---|

Volume | 2018-September |

ISSN (Print) | 1092-1648 |

### Conference

Conference | 26th International Conference on Network Protocols 2018 |
---|---|

Country/Territory | United Kingdom |

City | Cambridge |

Period | 25/09/18 → 27/09/18 |