Randomized Algorithms for Tracking Distributed Count, Frequencies, and Ranks

We show that randomization can lead to significant improvements for a few fundamental problems in distributed tracking. Our basis is the count-tracking problem, where there are $k$ players, each holding a counter $n_i$ that gets incremented over time, and the goal is to track an $\varepsilon$-approximation of their sum $n=\sum _i n_i$ continuously at all times, using minimum communication. While the deterministic communication complexity of the problem is ${\varTheta }(k/\varepsilon \cdot \log N)$, where $N$ is the final value of $n$ when the tracking finishes, we show that with randomization, the communication cost can be reduced to ${\varTheta }(k/\varepsilon \cdot \log N)$. Our algorithm is simple and uses only $O$(1) space at each player, while the lower bound holds even assuming each player has infinite computing power. Then, we extend our techniques to two related distributed tracking problems: frequency-tracking and rank-tracking, and obtain similar improvements over previous deterministic algorithms. Both problems are of central importance in large data monitoring and analysis, and have been extensively studied in the literature