## Abstract

We study the relation between latency and alphabet size in the context of Multicast Network Coding. Given a graph G = ({V}, E) representing a communication network, a subset S \subseteq V of sources, each of which initially holds a set of information messages, and a set T\subseteq V of terminals; we consider the problem in which one wishes to design a communication scheme that eventually allows all terminals to obtain all the messages held by the sources. In this study we assume that communication is performed in rounds, where in each round each network node may transmit a single (possibly encoded) information packet on any of its outgoing edges. The objective is to minimize the communication latency, i.e., number of communication rounds needed until all terminals have all the messages of the source nodes.For sufficiently large alphabet sizes (i.e., large block length, packet sizes), it is known that traditional linear multicast network coding techniques (such as random linear network coding)) minimize latency. In this work we seek to study the task of minimizing latency in the setting of limited alphabet sizes (\mathrm {i}.\mathrm {e}., finite block length), and alternatively, the task of minimizing the alphabet size in the setting of bounded latency. Through reductive arguments, we prove that it is NP-hard to (i) approximate (and in particular to determine) the minimum alphabet size given a latency constraint; (ii) to approximate (and in particular to determine) the minimum latency of communication schemes in the setting of limited alphabet sizes.

Original language | English |
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Title of host publication | 2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 212-218 |

ISBN (Electronic) | 9781538665961 |

DOIs | |

State | Published - 2 Jul 2018 |

Event | 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 - Monticello, United States Duration: 2 Oct 2018 → 5 Oct 2018 |

### Publication series

Name | 2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 |
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### Conference

Conference | 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 |
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Country/Territory | United States |

City | Monticello |

Period | 2/10/18 → 5/10/18 |

## Keywords

- Codes (symbols)
- Data communication systems
- Linear networks
- Multicasting
- Communication latency
- Communication rounds
- Communication schemes
- Finite block length
- Information messages
- Information packets
- Latency constraints
- Random Linear Network Coding
- Network coding