## Distance Oracles for Stretch Less Than 2 Abstract

### BibTeX

@MISC{Agarwal_distanceoracles,

author = {Rachit Agarwal},

title = {Distance Oracles for Stretch Less Than 2 Abstract},

year = {}

}

### OpenURL

### Abstract

We present distance oracles for weighted undirected graphs that return distances of stretch less than 2. For the realistic case of sparse graphs, our distance oracles exhibit a smooth three-way trade-off between space, stretch and query time — a phenomenon that does not occur in dense graphs. In particular, for any positive integer t and for any 1≤α ≤ n, our distance oracle is of size O(m+n 2 /α) and returns distances of stretch at most(1+ 2 t+1) in time O((αµ)t), whereµ=2m/n is the average degree of the graph. The query time can be further reduced to O((α+µ) t) at the expense of a small additive stretch. 1

### Citations

206 | Approximate distance oracles
- Thorup, Zwick
- 2005
(Show Context)
Citation Context ...size of the oracle and its stretch; this trade-off is now well understood for general undirected graphs. In particular, there exist distance oracles of sizeΘ(n 1.5 ) that return distances of stretch 3=-=[17]-=-, and a lower bound of Thorup and Zwick[17] shows that oracles that return distances of stretch less than 3 must have sizeΩ(n 2 ). However, the hard instances used to prove this lower bound are extrem... |

195 | Compact routing schemes
- Thorup, Zwick
- 2001
(Show Context)
Citation Context ...ples of the sparsity of a number of networks. These include nearly all networks with real-world applications of distance oracles — in social networks[1,13], personalized search[13,18], network routing=-=[2, 16]-=-, etc. In this sense, oracles that match the lower bound of[17] are optimal only for the obscure case of extremely dense graphs. The second reason sparse graphs are interesting is that the mathematica... |

121 | Analysis of topological characteristics of huge on-line social networking services - Ahn, Han, et al. - 2007 |

60 | All pairs shortest paths using bridging sets and rectangular matrix multiplication - Zwick |

52 |
The Cooperative Association for Internet Data Analysis. http://www.caida.org
- CAIDA
(Show Context)
Citation Context ... most relevant case. Nearly all large real-world networks are sparse, including road networks[14], social networks[4], the router-level Internet graph[9] and the Autonomous System-level Internet graph=-=[8]-=-, as well as networks like expander graphs that are important in many settings; see[2] 1 We say that a graph is sparse if it has m= o(n 2 ) edges. P. Brighten Godfrey Department of Computer Science, U... |

40 | Approximate distance oracles for unweighted graphs in expected O(n 2 ) time - Baswana, Sen |

25 | Fast shortest path distance estimation in large networks
- Potamias, Bonchi, et al.
- 2009
(Show Context)
Citation Context ...rbana-Champaign pbg@illinois.edu for numerical examples of the sparsity of a number of networks. These include nearly all networks with real-world applications of distance oracles — in social networks=-=[1,13]-=-, personalized search[13,18], network routing[2, 16], etc. In this sense, oracles that match the lower bound of[17] are optimal only for the obscure case of extremely dense graphs. The second reason s... |

20 | Faster algorithms for approximate distance oracles and all-pairs small stretch paths - Baswana, Kavitha - 2006 |

18 | Distance oracles for sparse graphs
- Sommer, Verbin, et al.
- 2009
(Show Context)
Citation Context ...acles that return distances of stretch 2 and 3 must requireΩ(n 2 ) andΩ(n 3/2 ) space. Their lower bounds, as discussed earlier, hold only for dense graphs and do not apply to our case. Sommer et al. =-=[15]-=- proved that the size of stretch-s time-t distance oracles is lower bounded by n 1+Ω(1/st) ; that is, for (constant stretch and) constant query time, any oracle must have superlinear size for graphs w... |

13 | Ramsey partitions and proximity data structures
- Mendel, Naor
(Show Context)
Citation Context ...size O(kn 1+1/k ) and returns stretch-(2k−1) distances in O(k) time; the construction time of their oracle is O(mn1/k ). Subsequent research improved the construction time[5–7, 19] and the query time =-=[10]-=-. Designing oracles with reduced size and/or stretch turned out to be a much harder problem, precisely due to the above lower bounds. Indeed, these results may be quite far from optimal for the realis... |

11 |
Distance oracles beyond the Thorup-Zwick bound
- Pǎtra¸scu, Roditty
- 2010
(Show Context)
Citation Context ...can be stored in relatively little space, and the trade-off with query time becomes critical. Relatively little is known about the space-stretchtime trade-off for sparse graphs. Pǎtra¸scu and Roddity =-=[11]-=- designed a stretch-2, constant-time oracle of size O(n 4/3 m 1/3 ). Agarwal et al.[2,3] explored the tradeoff between size and query time; their lowest-stretch oracle is of size O(m+nm 1−ǫ ) and retu... |

11 |
Route planning in road networks
- Schultes
- 2008
(Show Context)
Citation Context ...mportant for two reasons. First, far from being a narrow special case of the problem, sparse graphs are the most relevant case. Nearly all large real-world networks are sparse, including road networks=-=[14]-=-, social networks[4], the router-level Internet graph[9] and the Autonomous System-level Internet graph[8], as well as networks like expander graphs that are important in many settings; see[2] 1 We sa... |

11 | Efficient search ranking in social networks - Vieira, Fonseca, et al. - 2007 |

7 | Approximate distance queries and compact routing in sparse graphs
- Agarwal, Godfrey, et al.
- 2011
(Show Context)
Citation Context ...etworks[14], social networks[4], the router-level Internet graph[9] and the Autonomous System-level Internet graph[8], as well as networks like expander graphs that are important in many settings; see=-=[2]-=- 1 We say that a graph is sparse if it has m= o(n 2 ) edges. P. Brighten Godfrey Department of Computer Science, University of Illinois at Urbana-Champaign pbg@illinois.edu for numerical examples of t... |

5 | Distance oracles for unweighted graphs: Breaking the quadratic barrier with constant additive error - BASWANA, GAUR, et al. |

4 | Approximate distance oracles with improved preprocessing time - Wulff-Nilsen - 2012 |

2 | Shortest paths in less than a millisecond
- Agarwal, Caesar, et al.
- 2012
(Show Context)
Citation Context ...rbana-Champaign pbg@illinois.edu for numerical examples of the sparsity of a number of networks. These include nearly all networks with real-world applications of distance oracles — in social networks=-=[1,13]-=-, personalized search[13,18], network routing[2, 16], etc. In this sense, oracles that match the lower bound of[17] are optimal only for the obscure case of extremely dense graphs. The second reason s... |

2 |
The ipv4 routed/24 as links dataset
- Hyun, Huffaker, et al.
- 2010
(Show Context)
Citation Context ...special case of the problem, sparse graphs are the most relevant case. Nearly all large real-world networks are sparse, including road networks[14], social networks[4], the router-level Internet graph=-=[9]-=- and the Autonomous System-level Internet graph[8], as well as networks like expander graphs that are important in many settings; see[2] 1 We say that a graph is sparse if it has m= o(n 2 ) edges. P. ... |

1 | Faster approximate distance queries and compact routing in sparse graphs. http://arxiv.org/abs/1201.2703
- Agarwal, Godfrey, et al.
- 2012
(Show Context)
Citation Context ...arse cases. 3 Overview of our technique We start by giving a high level overview of our technique. To do so, let us briefly recall one of the most frequently used techniques to design distance oracles=-=[2, 3, 11, 17]-=-. Typically, the construction of distance oracles starts by selecting a subset of vertices L known as “landmark vertices”. The oracle stores the distance from each vertex in L to each other vertex in ... |