## Abstract

We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a sub linear-time algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries and neighbor queries. We show that for any given approximation parameter 0 <epsilon<1, the algorithm provides an estimate hat{t} such that with high constant probability, (1-epsilon) t<hat{t}<(1+epsilon)t, where t is the number of triangles in the graph G. The expected query complexity of the algorithm is O(n/t{1/3} + min {m, m {3/2}/t}) poly(log n, 1/epsilon), where n is the number of vertices in the graph and m is the number of edges, and the expected running time is (n/t{1/3} + m {3/2}/t) poly(log n, 1/epsilon). We also prove that Ω(n/t {1/3} + min {m, m {3/2}/t}) queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to polylogarithmic factors in n (and the dependence on 1/epsilon).

Original language | English |
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Title of host publication | Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 |

Publisher | IEEE Computer Society |

Pages | 614-633 |

Number of pages | 20 |

ISBN (Electronic) | 9781467381918 |

DOIs | |

State | Published - 11 Dec 2015 |

Externally published | Yes |

Event | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States Duration: 17 Oct 2015 → 20 Oct 2015 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2015-December |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 |
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Country/Territory | United States |

City | Berkeley |

Period | 17/10/15 → 20/10/15 |

### Bibliographical note

Publisher Copyright:© 2015 IEEE.

## Keywords

- Sublinear Approximation Algorithm
- Triangles Counting