Question

Given a tree, how to find the centre node in the tree so that the distance from the central node to other nodes is minimum(assuming each edge has unit weight)? I am trying to use DFS but is it possible to do it in linear time?

Answer 1

Keep removing leaf nodes from your tree until you are left with a single node (if left with two nodes, remove any one of them). That node minimizes the maximum distance from it to every other node.

Example:

   *                 *              
  / \                 \
 *   *                 *              *
      \                 \              \
       *      =>         *     =>       *    =>   *
        \                 \                     
         *                 *
          \
           *

To implement this in linear time, insert all initial leaf nodes in a FIFO queue. For each node, also store the number of its children. When removing an element from your queue, decrease its parent’s number of children. If this number becomes zero, insert the parent into the queue.