In the study of radio networks, the tasks of broadcasting (propagating a message throughout the network) and leader election (having the network agree on a node to designate `leader') are two of the most fundamental global problems, and have a long history of work devoted to them. This work has two divergent strands: some works focus on exploiting the geometric properties of wireless networks based in physical space, while others consider general graphs. Algorithmic results in each of these avenues have often used quite different techniques, and produced bounds using incomparable parametrizations. In this work, we unite the study of general-graph and geometric-based radio networks, by adapting the broadcast and leader election algorithm of Czumaj and Davies (JACM '21) to achieve a running-time parametrized by the independence number of the network (i.e., the size of the maximum independent set). This parametrization preserves the running time on general graphs, matching the best known, but also improves running times to near-optimality across a wide range of geometric-based graph classes. As part of this algorithm, we also provide the first algorithm for computing a maximal independent set in general-graph radio networks. This algorithm runs in $O(\log^3 n)$ time-steps, only a $\log n$ factor away from the $\Omega(\log^2 n)$ lower bound.

翻译：在无线网络研究中，广播（在网络中传播消息）和领导者选举（使网络就指定某个节点为"领导者"达成共识）是最基本的两个全局性问题，相关研究工作历史悠久。该研究存在两个分歧方向：部分工作致力于利用物理空间中无线网络的几何特性，而其他工作则考虑通用图。这两个方向上的算法结果通常采用截然不同的技术，并使用不可比较的参数化方法得出边界。在本工作中，我们通过改编Czumaj和Davies的广播与领导者选举算法（JACM '21），实现了以网络独立数（即最大独立集的大小）为参数化的运行时间，从而统一了通用图与几何无线网络的研究。该参数化在通用图上保持了与已知最优结果匹配的运行时间，同时将运行时间提升至接近最优水平，覆盖了广泛的几何图类。作为该算法的一部分，我们还首次提出了在通用图无线网络中计算最大独立集的算法。该算法运行时间为$O(\log^3 n)$时间步，仅比$\Omega(\log^2 n)$下界相差一个$\log n$因子。