# O(N) leader election on torus with orientation and non-positional identity

I'm really confused by this. Apparently there is a deterministic algorithm that does leader election on a torus with orientation and non-positional identity using only O(N) messages. I'm unable to find to find any description of this algorithm, and I still don't understand how it is even possible.

On a general graph with sense of direction you should be able to do leader election using O(N log N) messages. On complete graphs with sense of direction you do it with O(N) messages, because you have constant (1 message) communication with distant nodes. But how could you do that on a torus?

Here are some relevant details: The system is asynchronous. All processes start simultaneously. Each node has a "sense of direction"; that is, each node knows which of its neighbors are north, east, south, or west. Finally, each node has a unique ID (not related to its position in the torus).

• You should define your terms ("non-positional identity", "sense of direction", etc.) as well as the model of computation. Anyhow, if you are interested in synchronous deterministic distributed algorithms, and you have unique $O(\log n)$-bit identifiers, then it is trivial to elect a leader in $O(n)$ synchronous communication rounds on any connected graph: just find the smallest identifier. And if you happen care about the size of messages, $O(\log n)$-bit messages are sufficient. – Jukka Suomela Jan 20 '12 at 8:44
• @JukkaSuomela Clarified – Let_Me_Be Jan 20 '12 at 8:52
• Oh, I see, the challenge is to do it with $O(n)$ messages, not in $O(n)$ time? – Jukka Suomela Jan 20 '12 at 9:20
• Could you post a reference to the claim that such an algorithm exists? And do you already know how to solve the problem on a cycle? – Jukka Suomela Jan 20 '12 at 10:46
• (not a real answer but I can't write comments yet) I would be surprised if it exists. In an asynchronous cycle there's an \Omega(n\log n) lower bound on the average message complexity see Jan K. Pachl, Ephraim Korach, Doron Rotem: Lower Bounds for Distributed Maximum-Finding Algorithms. J. ACM 31(4): 905-918 (1984). – Peter Jan 20 '12 at 12:48

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 $O(n)$