### Abstract

Consider the following problem: We have k players each receiving a stream of items, and communicating with a central coordinator. Let the multiset of items received by player i up until time t be A_{i} (t). The coordinator's task is to monitor a given function f computed over the union of the inputs U_{i} A_{i} (t), continuously at all times t. The goal is to minimize the number of bits communicated between the players and the coordinator. Of interest is the approximate version where the coordinator outputs 1 if f ≥ τ and 0 if f ≤ (1-ε)τ . This defines the (k, f, τ, ε) 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 the well-studied streaming model and communication complexity. Yet few formal bounds are known for functional monitoring. We give upper and lower bounds for the (k, f, τ, ε) problem for some of the basic f's. In particular, we study the frequency moments F_{p} for p = 0, 1, 2. For F_{0} and F_{1}, we obtain monitoring algorithms with cost almost the same as algorithms that compute the function for a single instance of time. However, for F_{2} the monitoring problem seems to be much harder than computing the function for a single time instance. We give a carefully constructed multiround algorithm that uses "sketch summaries" at multiple levels of details and solves the (k, F_{2}, τ, ε) problem with communication Õ (k^{2}/ε + k^{3/2}/ε^{3}). Our algorithmic techniques are likely to be useful for other functional monitoring problems as well.

Original language | English (US) |
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Article number | 21 |

Journal | ACM Transactions on Algorithms |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2011 |

### Keywords

- Distributed computing
- Functional monitoring

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

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## Cite this

*ACM Transactions on Algorithms*,

*7*(2), [21]. https://doi.org/10.1145/1921659.1921667