## Abstract

Let k ≥ 2 be an integer. We show that any undirected and unweighted graph G = (V, E) on n vertices has a subgraph G′ = (V, E′) with O(kn^{1+1/k}) edges such that for any two vertices u, v ∈ V, if δ_{G}(u, v) = d, then δ_{G′}(u, v) = d + O(d^{1-1/k-1}). Furthermore, we show that such subgraphs can be constructed in O(mn^{1/k}) time, where m and n are the number of edges and vertices in the original graph. We also show that it is possible to construct a weighted graph G* = (V, E*) with O(kn ^{1+1/(2k-1)}) edges such that for every u, v ∈ V, if δ_{G}(u, v) = d, then d ≤ δ_{G}*(u, v) = d + O(d^{1-1/k-1}). These are the first such results with additive error terms. of the form o(d), i.e., additive error terms that are sublinear in the distance being approximated.

Original language | English |
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Pages | 802-809 |

Number of pages | 8 |

DOIs | |

State | Published - 2006 |

Event | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms - Miami, FL, United States Duration: 22 Jan 2006 → 24 Jan 2006 |

### Conference

Conference | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | Miami, FL |

Period | 22/01/06 → 24/01/06 |