Kth Smallest Element in a BST 解答

Question

Given a binary search tree, write a function kthSmallest to find the kth smallest element in it.

Note: 
You may assume k is always valid, 1 ≤ k ≤ BST‘s total elements.

Follow up

What if the BST is modified (insert/delete operations) often and you need to find the kth smallest frequently? How would you optimize the kthSmallest routine?

Hint:

1. Try to utilize the property of a BST.
2. What if you could modify the BST node‘s structure?
3. The optimal runtime complexity is O(height of BST).

Solution 1 -- Inorder Traversal

Again, we use the feature of inorder traversal of BST. But this solution is not best for follow up. Time complexity O(n), n is the number of nodes.

 1 /**
 2  * Definition for a binary tree node.
 3  * public class TreeNode {
 4  *     int val;
 5  *     TreeNode left;
 6  *     TreeNode right;
 7  *     TreeNode(int x) { val = x; }
 8  * }
 9  */
10 public class Solution {
11     public int kthSmallest(TreeNode root, int k) {
12         TreeNode current = root;
13         Stack<TreeNode> stack = new Stack<TreeNode>();
14         while (current != null || !stack.empty()) {
15             if (current != null) {
16                 stack.push(current);
17                 current = current.left;
18             } else {
19                 TreeNode tmp = stack.pop();
20                 k--;
21                 if (k == 0) {
22                     return tmp.val;
23                 }
24                 current = tmp.right;
25             }
26         }
27         return -1;
28     }
29 }

Solution 2 -- Augmented Tree

The idea is to maintain rank of each node. We can keep track of elements in a subtree of any node while building the tree. Since we need K-th smallest element, we can maintain number of elements of left subtree in every node.

Assume that the root is having N nodes in its left subtree. If K = N + 1, root is K-th node. If K < N, we will continue our search (recursion) for the Kth smallest element in the left subtree of root. If K > N + 1, we continue our search in the right subtree for the (K – N – 1)-th smallest element. Note that we need the count of elements in left subtree only.

Time complexity: O(h) where h is height of tree.

(referrence: GeeksforGeeks)

Here, we construct tree in a way that is taught during Algorithm class.

"size" is an attribute which indicates number of nodes in sub-tree rooted in that node.

Time complexity: constructing tree O(n), find Kth smallest number O(h).

start:
if K = root.leftElement + 1
   root node is the K th node.
   goto stop
else if K > root.leftElements
   K = K - (root.leftElements + 1)
   root = root.right
   goto start
else
   root = root.left
   goto srart

stop

 1 /**
 2  * Definition for a binary tree node.
 3  * public class TreeNode {
 4  *     int val;
 5  *     TreeNode left;
 6  *     TreeNode right;
 7  *     TreeNode(int x) { val = x; }
 8  * }
 9  */
10 class ImprovedTreeNode {
11     int val;
12     int size; // number of nodes in the subtree that rooted in this node
13     ImprovedTreeNode left;
14     ImprovedTreeNode right;
15     public ImprovedTreeNode(int value) {val = value;}
16 }
17
18 public class Solution {
19
20     // Construct ImprovedTree recursively
21     public ImprovedTreeNode createAugmentedBST(TreeNode root) {
22         if (root == null)
23             return null;
24         ImprovedTreeNode newHead = new ImprovedTreeNode(root.val);
25         ImprovedTreeNode left = createAugmentedBST(root.left);
26         ImprovedTreeNode right = createAugmentedBST(root.right);
27         newHead.size = 1;
28         if (left != null)
29             newHead.size += left.size;
30         if (right != null)
31             newHead.size += right.size;
32         newHead.left = left;
33         newHead.right = right;
34         return newHead;
35     }
36
37     public int findKthSmallest(ImprovedTreeNode root, int k) {
38         if (root == null)
39             return -1;
40         ImprovedTreeNode tmp = root;
41         int leftSize = 0;
42         if (tmp.left != null)
43             leftSize = tmp.left.size;
44         if (leftSize + 1 == k)
45             return root.val;
46         else if (leftSize + 1 > k)
47             return findKthSmallest(root.left, k);
48         else
49             return findKthSmallest(root.right, k - leftSize - 1);
50     }
51
52     public int kthSmallest(TreeNode root, int k) {
53         if (root == null)
54             return -1;
55         ImprovedTreeNode newRoot = createAugmentedBST(root);
56         return findKthSmallest(newRoot, k);
57     }
58 }