We present a self-stabilizing leader election algorithm for general networks, with space-complexity $O(log\Delta +loglogn)$ bits per node in $n$-node networks with maximum degree $\Delta$. This space complexity is sub-logarithmic in $n$ as long as $\Delta =n^{o\left(1\right)}$. The best space-complexity known so far for general networks was $O(logn)$ bits per node, and algorithms with sub-logarithmic space-complexities were known for the ring only. To our knowledge, our algorithm is the first algorithm for self-stabilizing leader election to break the $\Omega (logn)$ bound for silent algorithms in general networks. Breaking this bound was obtained via the design of a (non-silent) self-stabilizing algorithm using sophisticated tools such as solving the distance-2 coloring problem in a silent self-stabilizing manner, with space-complexity $O(log\Delta +loglogn)$ bits per node. Solving this latter coloring problem allows us to implement a sub-logarithmic encoding of spanning trees — storing the IDs of the neighbors requires $\Omega (logn)$ bits per node, while we encode spanning trees using $O(log\Delta +loglogn)$ bits per node. Moreover, we show how to construct such compactly encoded spanning trees without relying on variables encoding distances or number of nodes, as these two types of variables would also require $\Omega (logn)$ bits per node.

我们提出了一种具有空间复杂性的通用网络自稳定领导者选举算法 $Ø（日志\Delta +日志日志ñ）$ 每个节点中的位数 $ñ$度最大的节点网络 $\Delta$。这种空间复杂性是次对数的$ñ$ 只要 $\Delta ={ñ}^{Ø（\mathrm{1个}）}$。迄今为止，对于通用网络而言，最好的空间复杂度是$Ø（日志ñ）$每个节点只有1位，以及具有次对数空间复杂度的算法仅适用于环。据我们所知，我们的算法是第一个自我稳定的领导者选举打破僵局的算法。$\Omega （日志ñ）$限于一般网络中的静音算法。通过使用复杂工具设计（（非沉默）自稳定算法）可以突破这一界限，例如，以无声自稳定方式解决距离2着色问题，并具有空间复杂性$Ø（日志\Delta +日志日志ñ）$每个节点的位数。解决后一种着色问题，使我们能够实现生成树的亚对数编码-存储邻居的ID需要$\Omega （日志ñ）$ 每个节点的位数，而我们使用 $Ø（日志\Delta +日志日志ñ）$每个节点的位数。此外，我们展示了如何构建这种紧凑编码的生成树而不依赖于编码距离或节点数的变量，因为这两种类型的变量也需要$\Omega （日志ñ）$ 每个节点的位数。