## Abstract

The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the All-Pairs Shortest Paths (APSP) problem. In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] designed an algorithm that computes in Õ (n^{2} + m√n) time an estimate D̃ for the diameter D in directed graphs with nonnega- Tive edge weights, such that [2/3 · D]-(M - I) ≤ D̃ ≤ D, where M is the maximum edge weight in the graph. In recent work, Roditty and Vassilevska W. [STOC 13] gave a Las Vegas algorithm that has the same approximation guarantee but improves the (expected) runtime to Õ (m√;n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O(n^{2-ε}) time algorithm for sparse unweighted undirected graphs can achieve an approximation ratio better than 3/2. Thus their algorithm is essentially tight for sparse unweighted graphs. For weighted graphs however, the approximation guarantee can be meaningless, as M can be arbitrarily large. In this paper we exhibit two algorithms that achieve a genuine 3/2-approximation for the diameter, one running in Õ (m^{3/2})time, and one running in Õ (mn ^{2/3} ) time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 - ε)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs. In addition, we address the question of obtaining an additive c-approximation for the diameter, i.e. an estimate D̃ such that D - c ≤ D̃ ≤ D̃. An extremely simple Õ (mn^{1-ε}) time algorithm achieves an additive n ^{ε}- Approximation; no better results are known. We show that for any ε > 0, getting an additive n^{ε}-approximation algorithm for the diameter running in O (n^{2-δ}) time for any δ > 2ε would falsify the Strong Exponential Time Hypothesis. Thus the simple algorithm is probably essentially tight for sparse graphs, and moreover, obtaining a subquadratic time additive c-approximation for any constant c is unlikely. Finally, we consider the problem of computing the eccentricities of all vertices in an undirected graph, i.e. the largest distance from each vertex. Roditty and Vassilevska W. [STOC 13] show that in Õ(m√n) time, one can compute for each v ∈ V in an undirected graph, an estimate e (v) for the eccentricity ε (v) such that max {R, 2/3 · ε (v)} ≤ e (v) ≤ min {D, 3/2 · ε (v)} where R = min_{v}ε (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates ε' (v) with 3/5 · ε (v) ≤ ε' (v) ≤ ε(v).

Original language | English |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 1041-1052 |

Number of pages | 12 |

ISBN (Print) | 9781611973389 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: 5 Jan 2014 → 7 Jan 2014 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
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Country/Territory | United States |

City | Portland, OR |

Period | 5/01/14 → 7/01/14 |