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 ni\documentclass[12pt]{minimal}
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\begin{document}$$n_i$$\end{document} that gets incremented over time, and the goal is to track an ε\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon $$\end{document}-approximation of their sum n=∑ini\documentclass[12pt]{minimal}
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\begin{document}$$n=\sum _i n_i$$\end{document} continuously at all times, using minimum communication. While the deterministic communication complexity of the problem is Θ(k/ε·logN)\documentclass[12pt]{minimal}
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\begin{document}$${\varTheta }(k/\varepsilon \cdot \log N)$$\end{document}, where N is the final value of n when the tracking finishes, we show that with randomization, the communication cost can be reduced to Θ(k/ε·logN)\documentclass[12pt]{minimal}
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\begin{document}$${\varTheta }(\sqrt{k}/\varepsilon \cdot \log N)$$\end{document}. 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.
