## Abstract

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. We propose a series of new labeling schemes within the framework of so-called hub labeling (HL, also known as landmark labeling or 2-hop-cover labeling), in which each node u stores its distance to all nodes from an appropriately chosen set of hubs S(u) ⊆ V . For a queried pair of nodes (u; v), the length of a shortest u..v-path passing through a hub node from S(u)∩(v) is then used as an upper bound on the distance between u and v. We present a hub labeling which allows us to decode ex- act distances in sparse graphs using labels of size sublinear in the number of nodes. For graphs with at most n nodes and average degree δ, the tradeoff between label bit size L and query decoding time T for our approach is given by L = O(n log log_{δ} T= log_{δ} T), for any T ≤ n. Our simple approach is thus the first sublinear-space distance labeling for sparse graphs that simultaneously admits small decoding time (for constant δ, we can achieve any T = ω(1) while maintaining L = o(n)), and it also provides an improve- ment in terms of label size with respect to previous slower approaches. By using similar techniques, we then present a 2-additive labeling scheme for general graphs, i.e., one in which the decoder provides a 2-additive-approximation of the distance between any pair of nodes. We achieve the same label size- time tradeoff L = O(n log^{2} log T= log T), for any T ≤ n. To our knowledge, this is the first additive scheme with con- stant absolute error to use labels of sublinear size. The corresponding decoding time is then small (any T = ω(1) is sufficient).

Original language | English |
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Title of host publication | PODC 2016 - Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing |

Publisher | ACM |

Pages | 43-45 |

Number of pages | 3 |

Volume | 25-28-July-2016 |

ISBN (Electronic) | 9781450339643 |

DOIs | |

Publication status | Published - 25 Jul 2016 |

MoE publication type | A4 Article in a conference publication |

Event | ACM Symposium on Principles of Distributed Computing - Chicago, United States Duration: 25 Jul 2016 → 28 Jul 2016 Conference number: 35 |

### Conference

Conference | ACM Symposium on Principles of Distributed Computing |
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Abbreviated title | PODC |

Country | United States |

City | Chicago |

Period | 25/07/2016 → 28/07/2016 |

## Keywords

- Algorithms
- Distance Labeling
- Distance Oracles
- Distributed Data Structures
- Graphs