## Abstract

We study what we call functional monitoring problems. We have k players each tracking their inputs, say player i tracking a multiset A_{i}(t) up until time t, and communicating with a central coordinator. The coordinator's task is to monitor a given function / computed over the union of the inputs ∪_{i}A_{iv}(t), continuously at all times t. The goal is to minimize the number of bits communicated between the players and the coordinator. A simple example is when / is the sum, and the coordinator is required to alert when the sum of a distributed set of values exceeds a given threshold τ. Of interest is the approximate version where the coordinator outputs 1 if ≥ t and 0 if / ≤ (1 - ∈)τ. This defines the (k, f,τ,∈) distributed, functional monitoring problem. Functional monitoring problems are fundamental in distributed systems, in particular sensor networks, where we must minimize communication; they also connect to problems in communication complexity, communication theory, and signal processing. Yet few formal bounds are known lor functional monitoring. We give upper and lower bounds for the (k, f,τ,∈) problem for some of the basic f's. In particular, we study frequency moments (F_{0}, F_{1},F _{2}). For F_{0} and F_{1}, we obtain continuously monitoring algorithms with costs almost the same as their one-shot computation algorithms. However, for F_{2} the monitoring problem seems much harder. We give a carefully constructed multi-round algorithm that uses "sketch summaries" at multiple levels of detail and solves the (k, F _{2},τ, ∈) problem with communication Õ(k ^{2}/∈ + (√k/∈)^{3}). Since frequency moment estimation is central to other problems, our results have immediate applications to histograms, wavelet computations, and others. Our algorithmic techniques are likely to be useful for other functional monitoring problems as well.

Original language | English (US) |
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Title of host publication | Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms |

Publisher | Association for Computing Machinery |

Pages | 1076-1085 |

Number of pages | 10 |

ISBN (Print) | 9780898716474 |

State | Published - 2008 |

Event | 19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States Duration: Jan 20 2008 → Jan 22 2008 |

### Publication series

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

Other | 19th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | San Francisco, CA |

Period | 1/20/08 → 1/22/08 |

## ASJC Scopus subject areas

- Software
- General Mathematics