## Abstract

We study connection networks in which certain pairs of nodes have to be connected by k edge-disjoint paths, and study bounds for the minimal sum of lengths of such k paths. We define the related notions of total_{k}-distance for a pair of nodes and total_{k}-diameter of a connection network, and study the value TD_{k}(d) which is the maximal such total_{k}-diameter of a network with diameter d. These notions have applications in fault-tolerant routing problems, in ATM networks, and in compact routing in networks. We prove an upper bound on TD_{k}(d) and a lower bound on the growth of TD_{k}(d) as functions of k and d; those bounds are tight, θ(d^{k}), when k is fixed. Specifically, we prove that TD_{k}(d)≤2^{k-1}d^{k}, with the exceptions TD_{2}(1) = 3, TD_{3}(1) = 5, and that for every k,d_{0}>0, there exists (a) an integer d≥d_{0} such that TD_{k}(d)≥d^{k}/k^{k}, and (b) a k-connected simple graph G with diameter d such that d≥d_{0}, and whose total_{k}-diameter is at least (d - 2)^{k}/k^{k}.

Original language | English |
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Pages (from-to) | 213-228 |

Number of pages | 16 |

Journal | Theoretical Computer Science |

Volume | 247 |

Issue number | 1-2 |

DOIs | |

State | Published - 28 Sep 2000 |

Externally published | Yes |

## Keywords

- Communication network
- Connection network
- Edge-disjoint paths
- Flow in networks
- Total-diameter
- Total-distance

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