异步无向树中的领导者选举

Question

I have an asynchronous network undirected tree (V,E) with n=|V| processes. The only things that I know for my network is that all processes have unique ids (UIDs), they know the number of their neighbors but they do not know the diameter and the size of the network. I tried to construct a leader election algorithm in such a network as following:

  A convergecast of <leader> messages is initiated starting from the 
  leaves of the tree. 

  Each leaf node is initially enabled to send a <leader> message to 
  its unique neighbor. Any node that receives <leader> messages from 
  all but one of its neighbors is enabled to send an <leader> message 
  to its remaining neighbor.

  In the end,
    1. Some particular process receives <leader> messages along all
    of its channels before it has sent out an <leader> message
     the process at which the <leader> messages converge elects
    itself as the leader.

    2. <leader> messages are sent on some particular edge in both
    directions.
    the process with the largest pid among the processes that are
    adjacent to this edge elects itself as the leader.

Is my idea correct and does the above algorithm terminates with all the processes knowing the leader?

Answer 1

Your algorithm is correct, but has the following limitations:

1. Nodes only now their next leader, but not the global leader. I don't know whether your goal is that every node knows the global leader or their personal leader .
2. The global leader is not necessarily the process with the largest pid nor the root node of your tree. Example: Suppose a chain of 5 nodes with the second node as root. This graph is a tree with the first and the fifth node as leaves. Give the root node the largest pid. Your algorithm selects the third node of the chain as the global leader.
3. Your algorithm is in practice not deterministic: Suppose a balanced tree. Assume that one leave submits its leader message with a delay (can happen in practice). The selected global leader depends on the delay.
4. Your algorithm only works for trees.

I don't know whether these limitations are a problem for you. Since I don't have enough reputation to comment on your question, I'm also unable to ask.

The correctness of your algorithm is provable with mathematical induction.

1. Base case: Tree with a single node. Obviously, your algorithm is correct.

2. Induction step: Add a node to the tree. There are two cases:

   a) The new node is connected to a node which was previously not a leaf.

   b) The new node is connected to a node which was previously a leaf.

   It is not too difficult to prove that the output of the algorithm is correct in both cases. Thus the algorithm is correct for all trees.