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AVL Trees

AVL Trees. CSE, POSTECH. Balanced Binary Search Trees. If the height of a binary tree is always O(log n), we can guarantee O(log n) performance for each search tree operation Trees with a worst-case height of O(log n) are called balanced trees

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AVL Trees

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  1. AVL Trees CSE, POSTECH

  2. Balanced Binary Search Trees • If the height of a binary tree is always O(log n), we can guarantee O(log n) performance for each search tree operation • Trees with a worst-case height of O(log n) are called balanced trees • An example of a balanced tree is AVL (Adelson-Velsky and Landis) tree

  3. AVL Tree Definition • Binary tree. • If T is a nonempty binary tree with TL and TR as its left and right subtrees, then T is an AVL tree iff • TL and TR are AVL trees, and • |hL – hR|  1 where hL and hR are the heights of TL and TR, respectively

  4. AVL Search Trees • An AVL search tree is a binary search tree that is also an AVL tree • Which trees in Figure 14.1 are AVL trees? • (a) and (b) • Which trees in Figure 14.1 are AVL search trees? • (b) only • Which trees in Figure 14.3 are AVL search trees? • (a) and (b)

  5. Indexed AVL Search Trees • An indexed AVL search tree is an indexed binary search tree that is also an AVL tree • Which trees in Figure 14.2 are indexed AVL trees? • (a) and (b) • Which trees in Figure 14.2 are indexed AVL search trees? • (a) and (b)

  6. Properties of AVL Tree • The height of an AVL tree with n nodes is O(log n) • For every value of n, n 0, there exists an AVL tree • An n-node AVL search tree can be searched in O(height) = O(log n) time • A new node can be inserted into an n-node AVL search tree so that the result is an n+1 node AVL tree and insertion can be done in O(log n) time • A node can be deleted from an n-node AVL search tree, n>0, so that the result is an n-1 node AVL tree and deletion can be done in O(log n) time

  7. Balance Factor • AVL trees are normally represented using the linked representation • To facilitate insertion and deletion, a balance factor (bf) is associated with each node • The balance factor bf(x) of a node x is defined asheight(xleftChild) – height(xrightChild) • Balance factor of each node in an AVL tree must be –1, 0, or 1 • See Figure 15.1 for examples

  8. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 -1 0 1 5 20 35 60 0 25 AVL Tree with Balance Factors • Is this an AVL tree? • What is the balance factor for each node in this AVL tree? • Is this an AVL search tree?

  9. Searching an AVL Search Trees • To search an AVL search tree, we can use Program 14.4 (i.e., the code for searching a binary search tree) without any change • What would be the search time complexity? • O(log n)

  10. Inserting into an AVL Search Trees • If we use the strategy of Program 14.5 to insert an element into an AVL search tree, the result may not be an AVL tree • That is, the tree may become unbalanced • If the tree becomes unbalanced, we must adjust the tree to restore balance - this adjustment is called rotation • See the example in Figure 15.2 • Read the observations about the unbalanced tree that results from an insertion on pages 568-569

  11. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 -1 0 1 5 20 35 60 0 0 9 25 Inserting into an AVL Search Tree Insert(9) • Where is 9 going to be inserted into? • After the insertion, is the tree still an AVL search tree? (i.e., still balanced?)

  12. Imbalance Types • After an insertion, when the balance factor of node A is –2 or 2, the node A is one of the following four imbalance types • LL: new node is in the left subtree of the left subtree of A • LR: new node is in the right subtree of the left subtree of A • RR: new node is in the right subtree of the right subtree of A • RL: new node is in the left subtree of the right subtree of A

  13. Rotation Definition • To switch children and parents among two or three adjacent nodes to restore balance of a tree. • A rotation may change the depth of some nodes, but does not change their relative ordering.

  14. A 15 9 Left Rotation 22 4 9 15 12 12 4 22 Left Rotation Definition • In a binary search tree, pushing a node A down and to the left to balance the tree. • A's right child replaces A, and the right child's left child becomes A's right child. Animated rotation example: http://www.cs.queensu.ca/home/jstewart/applets/bst/bst-rotation.html

  15. A 9 15 Right Rotation 4 22 15 9 12 12 22 4 Right Rotation Definition • In a binary search tree, pushing a node A down and to the right to balance the tree. • A's left child replaces A, and the left child's right child becomes A's left child.

  16. AVL Rotations • To balance an unbalanced AVL tree (after an insertion), we may need to perform one of the following rotations: LL, RR, LR, RL Figure 15.3 Inserting into an AVL search tree

  17. Figure 15.4 An LL Rotation An LL Rotation

  18. Figure 15.5 An LR Rotation An LR Rotation

  19. Single and Double Rotations • Single rotations: the transformations done to correct LL and RR imbalances • Double rotations: the transformations done to correct LR and RL imbalances • The transformation to correct LR imbalance can be achieved by an RR rotation followed by an LL rotation • The transformation to correct RL imbalance can be achieved by an LL rotation followed by an RR rotation (do Exercise 15.13)

  20. Exercise 15.13 – LR Rotation • Figure (b) shows the tree after an RR rotation at node B • Figure (c) shows the result of performing an LL rotation at node A of the tree of Figure (b). • The resulting tree is the same as that shown in Figure 15.5 (c).

  21. AVL search tree insertion • See Figure 15.6 for the 4-step AVL-search-tree-insertion algorithm

  22. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 -1 0 1 5 20 35 60 0 25 29 Inserting into an AVL Search Tree Insert(29) • Where is 29 going to be inserted into? • - use the AVL-search-tree-insertion algorithm • in Figure 15.6) • After the insertion, is the tree still an AVL search tree? (i.e., still balanced?)

  23. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 0 1 5 20 35 60 -2 -1 25 0 29 Inserting into an AVL Search Tree • What are the new balance factors for 20, 25, 29? • What type of imbalance do we have? • RR imbalance  new node is in the right subtree of right subtree of node 20 (node with bf = -2)  what rotation do we need? • What would the left subtree of 30 look like after RR rotation?

  24. -1 10 1 1 7 40 0 0 1 -1 3 8 30 45 0 0 0 0 0 1 5 25 35 60 0 0 20 29 After RR Rotation • After the RR rotation, is the resulting tree an AVL search tree? • Do Exercise 15.1 • - see the solution on the Web http://www.cise.ufl.edu/~sahni/dsaac/

  25. Deletion from an AVL Search Tree • To delete an element from an AVL search tree, we can use Program 14.6 • Deletion of a node may also produce an imbalance • Imbalance incurred by deletion is classified intothe types R0, R1, R-1, L0, L1, and L-1 • Rotation is also needed for rebalancing • Read the observations after deleting a node from an AVL search tree • Read Section 15.1.6 for deletion from an AVL search tree

  26. An R0 Rotation Figure 15.7 An R0 rotation (single rotation)

  27. An R1 Rotation Figure 15.8 An R1 rotation (single rotation)

  28. Figure 15.9 An R-1 rotation (double rotation) An R-1 Rotation

  29. Exercise & Reading • Do Exercise 15.5 - see the solution on the Web http://www.cise.ufl.edu/~sahni/dsaac/ • READ Chapter 15.1

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