O(N) leader election on torus with orientation and non-positional identity
The idea is growing territories. That is, a candidate broadcast to neighbors that are $\alpha^i$ away in iterations. 

The trick is in the analysis! (so it s not really clear) -- by getting to know the worst case of candidates, and the number of messages they send + setting $\alpha$ to a certain constant - we get $O(n)$ 

The solution can be found in N. Santoro book in Chapter 3 (oriented torus)