The leader election task calls for all nodes of a network to agree on a
single node. If the nodes of the network are anonymous, the task of leader
election is formulated as follows: every node v of the network must output a
simple path, 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 anonymous trees.
Our aim is to establish tradeoffs between the allocated time τ and the
amount of information that has to be given a priori to the nodes to
enable leader election in time τ in all trees 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 tree. The length of this
string is called the size of advice. For an allocated time τ,
we give upper and lower bounds on the minimum size of advice sufficient to
perform leader election in time τ.
We consider n-node trees of diameter diam≤D. While leader election
in time diam can be performed without any advice, for time diam−1 we give
tight upper and lower bounds of Θ(logD). For time diam−2 we give
tight upper and lower bounds of Θ(logD) for even values of diam,
and tight upper and lower bounds of Θ(logn) for odd values of diam.
For the time interval [β⋅diam,diam−3] for constant β>1/2,
we prove an upper bound of O(Dnlogn) and a lower bound of
Ω(Dn), the latter being valid whenever diam is odd or when
the time is at most diam−4. Finally, for time α⋅diam for any
constant α<1/2 (except for the case of very small diameters), we give
tight upper and lower bounds of Θ(n)