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Lowest Common Ancestor of a Binary Tree Part I

July 18, 2011 in binary
tree

Given a binary tree, find the lowest common ancestor of two given nodes in the tree.

        _______3______
       /              \
    ___5__          ___1__
   /      \        /      \
   6      _2       0       8
         /  \
         7   4

If you are not so sure about the definition of lowest common ancestor (LCA), please refer to my previous post:Lowest
Common Ancestor of a Binary Search Tree (BST) or the definition of LCA here.
Using the tree above as an example, the LCA of nodes 5 and 1 is 3.
Please note that LCA for nodes 5 and 4 is 5.

Hint:

Top-down or bottom-up? Consider both approaches and see which one is more efficient.

A
Top-Down Approach (Worst case O(n²)
):

Let’s try the top-down approach where we traverse the nodes from the top to the bottom. First, if the current node is one of the two nodes, it must be the LCA of the two nodes. If not, we count the number of nodes that matches either p or q in the left subtree
(which we call totalMatches).
If totalMatches equals
1, then we know the right subtree will contain the other node. Therefore, the current node must be the LCA. If totalMatches equals
2, we know that both nodes are contained in the left subtree, so we traverse to its left child. Similar with the case wheretotalMatches equals
0 where we traverse to its right child.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

// 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); }

What is the run time complexity of this top-down approach?

First, just for fun, we assume that the tree contains n nodes
and is balanced (with its height equals to log(n)
). In this case, the run time complexity would be O(n).
Most people would guess a higher ordered complexity than O(n)
due to the function countMatchesPQ()
traverses the same nodes over and over again. Notice that the tree is balanced, you cut off half of the nodes you need to traverse in each recursive call of LCA()
function. The proof that the complexity is indeed O(n)
is left as an exercise to the reader.

What if the tree is not necessarily balanced? Then in the worst case the complexity could go up to O(n²).
Why? Could you come up with such case? (Hint: The tree might be a degenerate tree).

A
Bottom-up Approach (Worst case O(n)
):
Using a bottom-up approach, we can improve over the top-down approach by avoiding traversing the same nodes over and over again.

We traverse from the bottom, and once we reach a node which matches one of the two nodes, we pass it up to its parent. The parent would then test its left and right subtree if each contain one of the two nodes. If yes, then the parent must be the LCA and we
pass its parent up to the root. If not, we pass the lower node which contains either one of the two nodes (if the left or right subtree contains either p or q), or NULL (if both the left and right subtree does not contain either p or q) up.

Sounds complicated? Surprisingly the code appears to be much simpler than the top-down one.

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Node *LCA(Node *root, Node *p, Node *q) {   if (!root) return NULL;   if (root == p || root == q) return root;   Node *L = LCA(root->left, p, q);   Node *R = LCA(root->right, p, q);   if (L && R) return root;  // if p and q are on both sides   return L ? L : R;  // either one of p,q is on one side OR p,q is not in L&R subtrees }

Notes:
The LCA problem had been studied extensively by many computer scientists. There exists efficient algorithms for finding LCA in constant time after initial processing of the tree in linear time. For the adventurous reader, please read this article for
more details: Range
Minimum Query and Lowest Common Ancestor in Topcoder.

Further
Thoughts:

What if each node in the binary tree has a link to its parent? Could you devise a non-recursive approach without using extra space?

»
Continue reading Lowest
Common Ancestor of a Binary Tree Part II.

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Lowest Common Ancestor of a Binary Tree Part II

July 21, 2011 in binary
tree

        _______3______
       /              \
    ___5__          ___1__
   /      \        /      \
   6      _2       0       8
         /  \
         7   4