A Note on Leader Election Algorithms. Preliminary Report

In this paper we discuss a leader election algorithm which depends on two parameters $p, L$. The first one is a probability parameter of geometric distribution used during a draw of identity and the latter one is a maximal number of bits of memory that each node can develop in order to save identity of a potential leader. We consider a family $\{v_{1}, \ldots, v_{n}\}$ of nodes. In this algorithm each node $v_{i}$ generates independently a random number $x_{i}$ from a geometric distribution with parameter $p$ and furtherly calculates the number $y_{i}=\min(x_{i},\ L)$. Nodes which choose the biggest number become candidates for a leader. This procedure successfully elects the leader if there is exactly one candidate. We fix a number $N$ and $\varepsilon > 0$. Our goal is to determine such parameters $p$ and $L$ which guarantee that the considered algorithm will be accurate i.e. it will be successful with a probability at least $1 -\varepsilon$, for an arbitrary number of nodes $1\leq n\leq N$. This strong requirement distinguishes our considerations from many other analysis of leader election algorithms, which often focus only on theirs asymptotic correctness and properties. Our algorithm can be implemented either in single hop or in multi hop environment. In the single hop case it needs $\log_{2}L$ rounds to select a leader with required probability of success and may be adapted to the multi hop environment, where it needs $\mathrm{O}(d\log_{2}L)$ rounds, where $d$ denotes an upper bound for a diameter of the network.