We consider the Backup Placement problem in networks in the \(\mathcal {CONGEST}\) distributed setting. Given a network graph \(G = (V,E)\), the goal of each vertex \(v \in V\) is selecting a neighbor, such that the maximum number of vertices in V that select the same vertex is minimized. The backup placement problem was introduced by Halldorsson, Kohler, Patt-Shamir, and Rawitz, who obtained in 2015 an \(O(\log n/ \log \log n)\) approximation with randomized polylogarithmic time. Their algorithm remained state-of-the-art for general graphs, as well as for specific graph topologies. In the current paper, we obtain significantly improved algorithms for various graph topologies. Specifically, we show that O(1)-approximation to optimal backup placement can be computed deterministically in O(1) rounds (and even just one round) in wireless networks, certain social networks, claw-free graphs, and, more precisely, in any graph with neighborhood independence bounded by a constant. We also consider graphs such as trees, forests, planar graphs and, more precisely, graphs of constant arboricity. For such graphs, we obtain constant approximation to optimal backup placement in \(O(\log n)\) deterministic rounds. Clearly, our constant-time algorithms for graphs with constant neighborhood independence are asymptotically optimal. Moreover, we show that our algorithms for graphs with constant arboricity are not far from optimal as well by proving several lower bounds. Specifically, in unoriented trees, optimal backup placement requires \(\Omega (\log n)\) time and polylogarithmic-approximate backup placement requires \(\Omega (\sqrt{\log n / \log \log n})\) time. These lower bounds are applicable in particular to graphs of constant arboricity.

我们在\(\mathcal {CONGEST}\)分布式设置中考虑网络中的备份放置问题。给定一个网络图\(G = (V,E)\)，每个顶点\(v \in V\)的目标是选择一个邻居，使得V中选择相同顶点的最大顶点数最小化. 备份放置问题由 Halldorsson、Kohler、Patt-Shamir 和 Rawitz 提出，他们在 2015 年获得了\(O(\log n/ \log \log n)\)用随机多对数时间近似。他们的算法对于一般图以及特定图拓扑仍然是最先进的。在当前的论文中，我们为各种图拓扑获得了显着改进的算法。具体来说，我们表明，在无线网络、某些社交网络、无爪图，更准确地说，可以在O (1) 轮（甚至只是一轮）中确定性地计算最佳备份放置的O (1) 近似值，更准确地说，在任何邻域独立的图中，以常数为界. 我们还考虑了诸如树木、森林、平面图等图，更准确地说，是常乔木图。对于这样的图，我们在\(O(\log n)\)确定性轮次中获得最佳备份位置的恒定近似值。显然，我们对具有恒定邻域独立性的图的恒定时间算法是渐近最优的。此外，我们通过证明几个下界证明了我们的具有恒定树形性的图的算法也离最优不远。具体来说，在无向树中，最佳备份放置需要\(\Omega (\log n)\)时间，多对数近似备份放置需要\(\Omega (\sqrt{\log n / \log \log n})\)时间。这些下限尤其适用于恒定树形图。