How to compute a least common ancestor algorithm's time complexity?

Question

I came into an article which talking about the LCA algorithms, the code is simple http://leetcode.com/2011/07/lowest-common-ancestor-of-a-binary-tree-part-i.html

// Return #nodes that matches P or Q in the subtree.
int countMatchesPQ(Node *root, Node *p, Node *q) {
  if (!root) return 0;
  int matches = countMatchesPQ(root->left, p, q) + countMatchesPQ(root->right, p, q);
  if (root == p || root == q)
    return 1 + matches;
  else
    return matches;
}

Node *LCA(Node *root, Node *p, Node *q) {
  if (!root || !p || !q) return NULL;
  if (root == p || root == q) return root;
  int totalMatches = countMatchesPQ(root->left, p, q);
  if (totalMatches == 1)
    return root;
  else if (totalMatches == 2)
    return LCA(root->left, p, q);
  else /* totalMatches == 0 */
    return LCA(root->right, p, q);
}

but I was wondering how compute the time complexity of the algorithm,can anyone help me?

Answer 1

The worst case for this algorithm would be if the nodes are sibling leave nodes.

Node *LCA(Node *root, Node *p, Node *q)
{
  for root call countMatchesPQ;
  for(root->left_or_right_child) call countMatchesPQ; /* Recursive call */
  for(root->left_or_right_child->left_or_right_child) call countMatchesPQ;
  ...
  for(parent of leave nodes of p and q) call countMatchesPQ;
}

countMatchesPQ is called for height of tree times - 1. Lets call height of tree as h.

Now check the complexity of helper function

int countMatchesPQ(Node *root, Node *p, Node *q) {
  Search p and q in left sub tree recursively
  Search p and q in right sub tree recursively
}

So this is an extensive search and the final complexity is N where N is the number of nodes in the tree.

Adding both observations, total complexity of the algorithm is

O(h * N)

If tree is balanced, h = log N (RB tree, treap etc) If tree is unbalanced, in worse case h may be up to N

So complexity in terms of N can be given as

For balanced binary tree: O(N logN)
To be more precise, it is actual h(N + N/2 + N/4...) for balanced tree and hence should come 2hN
For unbalanced binary tree: O(N²)
To be more precise, it is actual h(N + N-1 + N-2...) for balanced tree and hence should come h x N x (N+1) / 2

So the worse case complexity is N²

Your algorithm doesn't use any memory. By using some memory to save path, you can improve your algorithm drastically.

Answer 2

The complexity of LCA is O(h) where h is the height of the tree. The upper bound of the tree height is O(n), where n denotes the number of vertices/nodes in the tree.

If your tree is balanced, (see AVL, red black tree) the height is order of log(n), consequently the total complexity of the algorithm is O(log(n)).