In 2013 P. Jacquet et all (see   : 1 ) proposed a new algorithm for leader election which can work in a single-hop ad-hoc network. He called this algorithm the Green Leader Election (GLE) algorithm. We describe now a brief description of this algorithm. All devices will run the same algorithm. This algoritm has two parameters:

1. \(p\) - some probability,
2. \(N\) - upper bound on the number of devices.

\[ L = \max\{2 \left\lceil\log_3\left( \frac{\ln N}{\ln(\frac{1}{1-p})} \right)\right\rceil, 2\} \] This is the initial step of this algorithm:

1. select a random natural number k according to the geometric distribution with parameter p
2. express k as \( k = \sum_{i=0}^{L} a_i 3^i \)
3. construct a sequence s = \( f(a_L) || f(a_{L-1})||\ldots || f(a_0) \), where f(0)='00', f(1) = '01', f(2)='10' and \( || \) denotes the concatenation of sequences

LEADER = true;
FOR I=1 TO length(s) DO
  IF (s[I]==1) {
    send message;
  } ELSE {
    listen;
	if (hear message or noise){
	  LEADER = false;
	  BREAK;
	}
  }
OD

The crucial point in analysis of this algorithm is based on the following fact: if \(X_1\), ... , \(X_m\) are pairwise independent random variables with common geometric distribution with parameter \(p\) and \(Y = \max\{X_1, \ldots, X_m\}\) then \[E(Y) \approx \frac{\ln N}{\ln(\frac{1}{1-p})} ~,\] where \(E(Y) \) denotes the expected value of the random variable \(Y\) (see   : 2 ).

It can be proved (however the formal analysis of this fact is quite complicated) that after the end of this algorithm with probability \( \approx 1 - \frac{1}{2} p\) there will be only one device with the variable LEADER set to true (independently to the number of devices !!!).

Notice that the maximum number of times when the given device sends a message is bouded by the number L, hence total energy cost is of order \( O(\log\log(N)) \).

However, we propose an different approach, namely we change a strategy after the first step. But firstly we need one auxiliary result.

Let \( L_{p,N} \) be the number of devices with the variable LEADER set to true at the end of GLE algorithm. Notice that \( L_{p,N} \) is a random variable. A careful and quite complicated analysis of the LTE algorithm gives a possibility of estimating the number \( \Pr[L_{p,N} \leq k] \) for a given number k. This formulas shows that \( \Pr[L_{10^{-2},N} \gt 10] \approx 10^{-20} \). Hence with a very high probability we may expect that the number \( L_{p,N} \) is bounded by 10.