## Abstract

Given a weighted undirected graph, our basic goal is to represent all pair wise distances using much less than quadratic space, such that we can estimate the distance between query vertices in constant time. We will study the inherent trade-off between space of the representation and the stretch (multiplicative approximation disallowing underestimates) of the estimates when the input graph is sparse with m= Õ(n) edges. In this paper, for any fixed positive integers k and ℓ, we obtain stretches α=2k+1±2/ℓ = 2k+1-2/ℓ, 2k+1+2/ℓ, using space S(α, m) = Õ(m ^{1+2/(α+1)}). The query time is O(k+ℓ)=O(1). For integer stretches, this coincides with the previous bounds (odd stretches with ℓ=1 and even stretches with ℓ=2). The infinity of fractional stretches between consecutive integers are all new (even though ℓ is fixed as a constant independent of the input, the number of integers ℓ is still countably infinite). % We will argue that the new fractional points are not just arbitrary, but that they, at least for fixed stretches below 3, provide a complete picture of the inherent trade-off between stretch and space in m. Consider any fixed stretch α < 3. Based on the hardness of set intersection, we argue that if ℓ is the largest integer such that 3-2/ℓ ≤ α, then Ω(S(3 - 2/ℓ, m)) space is needed for stretch α. In particular, for ?xed stretch below 2 2/3, we improve Pǎtraşcu and Roditty's lower bound from Ω(m^{3/2}) to Ω(m^{5/3}), thus matching their upper bound for stretch 2. For space in terms of m, this is the first hardness matching the space of a non-trivial/sub-quadratic distance oracle.

Original language | English |
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Article number | 6375353 |

Pages (from-to) | 738-747 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 2012 |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: 20 Oct 2012 → 23 Oct 2012 |

## Keywords

- distance oracles
- distances
- shortest paths
- sparse graphs

## All Science Journal Classification (ASJC) codes

- General Computer Science