Leader election is one of the fundamental problems in distributed computing: a single node, called the leader, must be specified. This task can be formulated either in a weak way, where one node outputs leader and all other nodes output non-leader, or in a strong way, where all nodes must also learn which node is the leader. If the nodes of the network have distinct identifiers, then such an agreement means that all nodes have to output the identifier of the elected leader. For anonymous networks, the strong version of leader election requires that all nodes must be able to find a path to the leader, as this is the only way to identify it. In this paper, we study variants of deterministic leader election in arbitrary anonymous networks. Leader election is impossible in some anonymous networks, regardless of the allocated amount of time, even if nodes know the entire map of the network. This is due to possible symmetries in the network. However, even in networks in which it is possible to elect a leader knowing the map, the task may be still impossible without any initial knowledge, regardless of the allocated time. On the other hand, for any network in which leader election (weak or strong) is possible knowing the map, there is a minimum time, called the election index, in which this can be done. We consider four formulations of leader election discussed in the literature in the context of anonymous networks: one is the weak formulation, and the three others specify three different ways of finding the path to the leader in the strong formulation. Our aim is to compare the amount of initial information needed to accomplish each of these “four shades” of leader election in minimum time. Following the framework of algorithms with advice, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire network. The length of this string is called the size of advice. We show that the size of advice required to accomplish leader election in the weak formulation in minimum time is exponentially smaller than that needed for any of the strong formulations. Thus, if the required amount of advice is used as a measure of the difficulty of the task, the weakest version of leader election in minimum time is drastically easier than any version of the strong formulation in minimum time.

领导者选举是分布式计算中的基本问题之一：必须指定称为领导者的单个节点。这个任务可以用一种弱方式来表述，其中一个节点输出领导者，所有其他节点输出非领导者，或者以一种强方式，所有节点还必须了解哪个节点是领导者。如果网络的节点具有不同的标识符，那么这样的协议意味着所有节点都必须输出当选领导者的标识符。对于匿名网络，强版本的领导者选举要求所有节点必须能够找到通往领导者的路径，因为这是识别领导者的唯一方法。在本文中，我们研究了任意匿名网络中确定性领导人选举的变体。在某些匿名网络中，无论分配的时间长短如何，即使节点知道网络的整个地图，领导者选举也是不可能的。这是由于网络中可能存在的对称性。然而，即使在可以选择知道地图的领导者的网络中，如果没有任何初始知识，任务仍然可能是不可能的，不管分配的时间。另一方面，对于可能知道地图的领导人选举（弱或强）的任何网络，都有一个最短时间，称为选举指数，这可以在其中完成。我们考虑在匿名网络的背景下文献中讨论的四种领导人选举公式：一种是弱公式，另外三种指定三种不同的方法来找到强公式中的领导者路径。我们的目标是比较在最短时间内完成领导人选举的“四种阴影”中的每一种所需的初始信息量。遵循带有建议的算法框架，此信息（单个二进制字符串）在开始时由了解整个网络的预言机提供给所有节点。这个字符串的长度称为建议的大小. 我们表明，在最短时间内完成弱公式中的领导者选举所需的建议规模比任何强公式所需的建议规模都小得多。因此，如果将所需的建议量用作任务难度的衡量标准，则最短时间内领导者选举的最弱版本比最短时间内的任何强版本都要容易得多。