## Abstract

We present a uniform approach to derive message-time tradeoffs and message lower bounds for synchronous distributed computations using results from communication complexity theory. Since the models used in the classical theory of communication complexity are inherently asynchronous, lower bounds do not directly apply in a synchronous setting. To address this issue, we show a general result called Synchronous Simulation Theorem (SST) which allows to obtain message lower bounds for synchronous distributed computations by leveraging lower bounds on communication complexity. The SST is a by-product of a new efficient synchronizer for complete networks, called σ, which has simulation overheads that are only logarithmic in the number of synchronous rounds with respect to both time and message complexity in the CONGEST model. The σ synchronizer is particularly efficient in simulating synchronous algorithms that employ silence. In particular, a curious property of this synchronizer, which sets it apart from its predecessors, is that it is time-compressing, and hence in some cases it may result in a simulation that is faster than the original execution. While the SST gives near-optimal message lower bounds up to large values of the number of allowed synchronous rounds r (usually polynomial in the size of the network), it fails to provide meaningful bounds when a very large number of rounds is allowed. To complement the bounds provided by the SST, we then derive message lower bounds for the synchronous message-passing model that are unconditional, that is, independent of r, via direct reductions from multi-party communication complexity. We apply our approach to show (almost) tight message-time tradeoffs and message lower bounds for several fundamental problems in the synchronous message-passing model of distributed computation. These include sorting, matrix multiplication, and many graph problems. All these lower bounds hold for any distributed algorithms, including randomized Monte Carlo algorithms.

Original language | English |
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Title of host publication | Structural Information and Communication Complexity - 23rd International Colloquium, SIROCCO 2016, Revised Selected Papers |

Editors | Jukka Suomela |

Publisher | Springer Verlag |

Pages | 75-91 |

Number of pages | 17 |

ISBN (Print) | 9783319483139 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 23rd International Colloquium on Structural Information and Communication Complexity, SIROCCO 2016 - Helsinki, Finland Duration: 19 Jul 2016 → 21 Jul 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9988 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 23rd International Colloquium on Structural Information and Communication Complexity, SIROCCO 2016 |
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Country/Territory | Finland |

City | Helsinki |

Period | 19/07/16 → 21/07/16 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing AG 2016.

### Funding

G. Pandurangan—Supported, in part, by NSF grants CCF-1527867 and CCF-1540512.

Funders | Funder number |
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National Science Foundation | CCF-1527867, CCF-1540512 |