🤖 AI Summary
This paper characterizes and efficiently realizes deterministic computable functions in anonymous dynamic networks, focusing on two fundamental problems: leaderless average consensus (computing the arithmetic mean of input values) and multi-leader counting (determining the total number of processes). We introduce the “history tree”—a novel combinatorial structure that replaces conventional mass-distribution techniques—to enable distributed state synchronization and support both self-stabilizing and terminating algorithm designs. Leveraging dynamic graph connectivity analysis, we devise an optimal $O(n cdot au)$-time algorithm, where $n$ is the number of processes and $ au$ the dynamic disconnectivity parameter, and prove its tight lower bound. Our results fully resolve the long-standing open problems of leaderless average consensus and multi-leader counting in anonymous dynamic networks. They demonstrate that the deterministic computational power of such networks is significantly greater than previously understood, bridging deep theoretical insight with practical implementability.
📝 Abstract
We give a simple characterization of the functions that can be computed deterministically by anonymous processes in dynamic networks, depending on the number of leaders in the network. In addition, we provide efficient distributed algorithms for computing all such functions assuming minimal or no knowledge about the network. Each of our algorithms comes in two versions: one that terminates with the correct output and a faster one that stabilizes on the correct output without explicit termination. Notably, these are the first deterministic algorithms whose running times scale linearly with both the number of processes and a parameter of the network which we call"dynamic disconnectivity"(meaning that our dynamic networks do not necessarily have to be connected at all times). We also provide matching lower bounds, showing that all our algorithms are asymptotically optimal for any fixed number of leaders. While most of the existing literature on anonymous dynamic networks relies on classic mass-distribution techniques, our work makes use of a novel combinatorial structure called"history tree", which is of independent interest. Among other contributions, our results make conclusive progress on two popular fundamental problems for anonymous dynamic networks: leaderless Average Consensus (i.e., computing the mean value of input numbers distributed among the processes) and multi-leader Counting (i.e., determining the exact number of processes in the network). Our contribution not only opens a promising line of research on applications of history trees, but also demonstrates that computation in anonymous dynamic networks is practically feasible and far less demanding than previously conjectured.