Balanced Binary Search Tree

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1 AVL Trees / Slide 1 Balanced Binary Search Tree Worst case height of binary search tree: N-1 Insertion, deletion can be O(N) in the worst case We want a tree with small height Height of a binary tree with N node is at least O(log N) Complete binary tree has height log N. Goal: keep the height of a binary search tree O(log N) Balanced binary search trees Examples: AVL tree, red-black tree

2 AVL Trees / Slide 2 AVL tree for each node, the height of the left and right subtrees can differ by at most d = 1. Maintaining a stricter condition (e.g above condition with d = 0) is difficult. Note that our goal is to perform all the operations search, insert and delete in O(log N) time, including the operations involved in adjusting the tree to maintain the above balance condition.

3 AVL Trees / Slide 3 AVL Tree An AVL tree is a binary search tree in which for every node in the tree, the height of the left and right subtrees differ by at most 1. Height of subtree: Max # of edges to a leaf Height of an empty subtree: -1 Height of one node: 0 AVL tree AVL tree property violated here

4 AVL Trees / Slide 4 AVL Tree with Minimum Number of Nodes N 0 = 1 N 1 = 2 N 2 =4 N 3 = N 1 +N 2 +1=7

5 AVL Trees / Slide 5 Smallest AVL tree of height 7 Smallest AVL tree of height 8 Smallest AVL tree of height 9

6 AVL Trees / Slide 6 Height of AVL Tree with N nodes Denote N h the minimum number of nodes in an AVL tree of height h N 0 = 1, N 1 =2 (base) N h = N h-1 + N h-2 +1 (recursive relation) N > N h = N h-1 + N h-2 +1 > 2 N h-2 > 4 N h-4 >... >2 i N h-2i

7 AVL Trees / Slide 7 Height of AVL Tree with N nodes If h is even, let i=h/2 1. The equation becomes N>2 h/2-1 N N > 2 2h/2-1 x4 h=o(log N) If h is odd, let i=(h-1)/2. The equation becomes N>2 (h-1)/2 N 1 N > 2 (h-1)/2 x2 h=o(log N) Recall the cost of operations search, insert and delete is O(h). cost of searching = O(log N). (Just use the search algorithm for the binary search tree.) But insert and delete are more complicated.

8 AVL Trees / Slide 8 Insertion in AVL Tree Basically follows insertion strategy of binary search tree But may cause violation of AVL tree property Restore the destroyed balance condition if needed Original AVL tree 6 Insert 6 Property violated Restore AVL property

9 AVL Trees / Slide 9 Some Observations After an insertion (using the insertion algorithm for a binary search tree), only nodes that are on the path from the insertion point to the root might have their balance altered. Because only those nodes have their subtrees altered So it s enough to adjust the balance on these nodes. We will see that AVL tree property can be restored by performing a balancing operation at a single node.

10 AVL Trees / Slide 10 Node at which balancing must be done The node of smallest depth on the path from to the newly inserted node. We will call this node pivot. Example: Suppose we inserted key is 2.5 * Pivot is the node containing 4. This is the lowest depth node in the path where the AVL tree property is violated. 2.5

11 AVL Trees / Slide 11 Different Cases for Rebalance Let α be the pivot node Case 1: inserted node is in the left subtree of the left child of α (LL-rotation) Case 2: inserted node is in the right subtree of the left child of α (RL-rotation) Case 3: inserted node is in the left subtree of the right child of α (LR-rotation) Case 4: inserted node is in the right subtree of the right child of α (RR-rotation) Cases 3 & 4 are mirror images of cases 1 & 2 so we will focus on 1 & 2.

12 AVL Trees / Slide 12 Rotations Rebalance of AVL tree are done with simple modification to tree, known as rotation Insertion occurs on the outside (i.e., left-left or right-right) is fixed by single rotation of the tree Insertion occurs on the inside (i.e., left-right or right-left) is fixed by double rotation of the tree

13 AVL Trees / Slide 13 Insertion Algorithm (outline) First, insert the new key as a new leaf just as in ordinary binary search tree Then trace the path from the new leaf towards the root. For each node x encountered, check if heights of left(x) and right(x) differ by at most 1. If yes, proceed to parent(x) If not, restructure by doing either a single rotation or a double rotation Note: once we perform a rotation at a node x, we won t need to perform any rotation at any ancestor of x.

14 AVL Trees / Slide 14 Single Rotation to Fix Case 1(LL-rotation) k2 is the pivot node Questions: Can Y have the same height as the new X? Can Y have the same height as Z?

15 AVL Trees / Slide 15 Single Rotation Case 1 Example k2 k1 k1 X k2 X

16 AVL Trees / Slide 16 Informal proof/argument for the LL-case We need to show the following: After the rotation, the balance condition at the pivot node is restored and at all the nodes in the subtree rooted at the pivot. After the rotation, the balance condition is restored at all the ancestors of the pivot. We will prove both these informally.

17 AVL Trees / Slide 17 Single Rotation to Fix Case 4 (RR-rotation) k1 violates AVL tree balance condition An insertion in subtree Z Case 4 is a symmetric case to case 1 Insertion takes O(Height of AVL Tree) time, Single rotation takes O(1) time

18 AVL Trees / Slide 18 Single Rotation Example Sequentially insert 3, 2, 1, 4, 5, 6 to an AVL Tree 2 3 Insert 3, Single rotation Single rotation Insert 1 violation at node Insert 4 5 Insert 6, violation at node Insert 5, 4 violation at node Single rotation

19 AVL Trees / Slide 19 Single Rotation Fails to fix Case 2&3 Case 2: violation in k2 because of insertion in subtree Y Single rotation result Single rotation fails to fix case 2&3 Take case 2 as an example (case 3 is a symmetry to it ) The problem is subtree Y is too deep Single rotation doesn t make it any less deep

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