Question

I’m trying to find a linear-time algorithm using recursion to solve the diameter problem for a rooted k-ary tree implemented with adjacency lists. The diameter of a tree is the maximum distance between any couple of leaves. If I choose a root r (that is, a node whose degree is > 1), it can be shown that the diameter is either the maximum distance between two leaves in the same subtree or the maximum distance between two leaves of a path that go through r. My pseudocode for this problem:

Tree-Diameter(T,r)
    if degree[r] = 1 then
        height[r] = 0
        return 0
    for each v in Adj[r] do
        for i = 1 to degree[r] - 1 do
            d_i = Tree-Diameter(T,v)
    height[r] = max_{v in Adj[r]} (height[v]
    return max(d_i, max_{v in V} (height[v]) + secmax_{v in V} (height[v], 0) + 1)

To get linear time, I compute the diameter AND the height of each subtree at the same time. Then, I choose the maximum quantity between the diameters of each subtrees and the the two biggest heights of the tree + 1 (the secmax function chooses between height[v] and 0 because some subtree can have only a child: in this case, the second biggest height is 0). I ask you if this algorithm works fine and if not, what are the problems? I tried to generalize an algorithm that solve the same problem for a binary tree but I don’t know if it’s a good generalization.

Any help is appreciated! Thanks in advance!

Answer 1

This is a python implementation of what I believe you are interested in. Here, a tree is represented as a list of child trees.

def process(tree):
  max_child_height=0
  secmax_child_height=0
  max_child_diameter=0
  for child in tree:
    child_height,child_diameter=process(child)
    if child_height>max_child_height:
      secmax_child_height=max_child_height
      max_child_height=child_height
    elif child_height>secmax_child_height:
      secmax_child_height=child_height
    if child_diameter>max_child_diameter:
      max_child_diameter=child_diameter
  height=max_child_height+1
  if len(tree)>1:
    diameter=max(max_child_diameter,max_child_height+secmax_child_height)
  else:
    diameter=max_child_diameter
  return height,diameter

def diameter(tree):
  height,diameter=process(tree)
  return diameter