Impact of Knowledge on Election Time in Anonymous Networks

Abstract

Leader election is one of the basic problems in distributed computing. This is a symmetry breaking problem: all nodes of a network must agree on a single node, called the leader. If the nodes of the network have distinct labels, then such an agreement means that all nodes have to output the label of the elected leader. For anonymous networks, the task of leader election is formulated as follows: every node v of the network must output a simple path, which is coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this paper, we study deterministic leader election in arbitrary anonymous networks. It is well known that deterministic leader election is impossible in some networks, regardless of the allocated amount of time, even if nodes know the map of the network. This is due to possible symmetries in it. However, even in networks in which it is possible to elect a leader knowing the map, the task may be still impossible without any knowledge, regardless of the allocated time. On the other hand, for any network in which leader election is possible knowing the map, there is a minimum time, called the election index, in which this can be done. Informally, the election index of a network is the minimum depth at which views of all nodes are distinct. Our aim is to establish tradeoffs between the allocated time \tau and the amount of information that has to be given a priori to the nodes to enable leader election in time \tau in all networks for which leader election in this time is at all possible. 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. For a given time \tau allocated to leader election, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time \tau. We focus on the two sides of the time spectrum. For the smallest possible time, which is the election index of the network, we show that the minimum size of advice is linear in the size n of the network, up to polylogarithmic factors. On the other hand, we consider large values of time: larger than the diameter D by a summand, respectively, linear, polynomial, and exponential in the election index; for these values, we prove tight bounds on the minimum size of advice, up to multiplicative constants. We also show that constant advice is not sufficient for leader election in all graphs, regardless of the allocated time.