## Abstract

We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to solve a graph problem. We show that the local information cost (LIC) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT_{1} model, where each node has initial knowledge of its neighbors' IDs, we prove that Ω ( _{log}^{LIC}_{τ}^{γ}_{log}^{(}P)_{n)} bits are required for solving a graph problem P with a τ-round algorithm that errs with probability at most γ. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT_{1} model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a multiplicative spanner with stretch 2t − 1 that consists of at most O(n^{1+ 1 t +ε}) edges, where ε = O(1/t^{2)}. Our main result is that any O(poly(n))-time algorithm must send at least Ω-( _{t}^{1}_{2} n^{1+1}/2^{t)} bits in the CONGEST model under the KT_{1} assumption. Previously, only a trivial lower bound of Ω(-n) bits was known for this problem; in fact, this is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem in this setting. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with Õ(n) communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.

Original language | English (US) |
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Title of host publication | ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 |

Editors | Daniel Marx |

Publisher | Association for Computing Machinery |

Pages | 2105-2120 |

Number of pages | 16 |

ISBN (Electronic) | 9781611976465 |

State | Published - 2021 |

Externally published | Yes |

Event | 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States Duration: Jan 10 2021 → Jan 13 2021 |

### Publication series

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

Conference | 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 |
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Country/Territory | United States |

City | Alexandria, Virtual |

Period | 1/10/21 → 1/13/21 |

## ASJC Scopus subject areas

- Software
- Mathematics(all)