## Abstract

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading: Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of Θ(n) for bounded degree graphs and an upper bound of O(n^{2}) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research.

Original language | English |
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Pages (from-to) | 185-217 |

Number of pages | 33 |

Journal | Random Structures and Algorithms |

Volume | 45 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2014 |

## Keywords

- Algebraic gossip
- Gossip
- Gossip algorithms
- Network capacity
- Network coding

## ASJC Scopus subject areas

- Software
- Mathematics (all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics