# Monitoring of Domain-Related Problems in Distributed Data Streams

## TL;DR

This paper develops randomized algorithms for monitoring domain-related problems in distributed data streams, optimizing communication while accurately tracking data set properties over time.

## Contribution

It introduces novel algorithms for efficient monitoring of data domains, sizes, and frequencies in distributed streams, with improved bounds under realistic input assumptions.

## Key findings

- Algorithms achieve reduced communication costs.
- Improved bounds when data changes are limited.
- Effective monitoring of domain size and element frequencies.

## Abstract

Consider a network in which $n$ distributed nodes are connected to a single server. Each node continuously observes a data stream consisting of one value per discrete time step. The server has to continuously monitor a given parameter defined over all information available at the distributed nodes. That is, in any time step $t$, it has to compute an output based on all values currently observed across all streams. To do so, nodes can send messages to the server and the server can broadcast messages to the nodes. The objective is the minimisation of communication while allowing the server to compute the desired output. We consider monitoring problems related to the domain $D_t$ defined to be the set of values observed by at least one node at time $t$. We provide randomised algorithms for monitoring $D_t$, (approximations of) the size $|D_t|$ and the frequencies of all members of $D_t$. Besides worst-case bounds, we also obtain improved results when inputs are parameterised according to the similarity of observations between consecutive time steps. This parameterisation allows to exclude inputs with rapid and heavy changes, which usually lead to the worst-case bounds but might be rather artificial in certain scenarios.