Data Structure and Algorithms AVL Trees

Data Structure and Algorithms AVL Trees

• 1.
  Introduction Insertion Deletion
• 2.
  Balance Binary SearchTree Worst case height of binary search tree: N 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 Θ(log N) Goal: keep the height of a binary search tree O(log N) Balanced binary search trees Examples: AVL tree, red-black tree
• 3.
  Balanced Tree? Suggestion 1:the left and right subtrees of root have the same height Suggestion 2: every node must have left and right subtrees of the same height Our choice: for each node, the height of the left and right subtrees can differ at most 1,-1,0.
• 4.
  AVL Tree An AVL(Adelson-Velskii and Landis 1962) 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. AVL property violated here AVL tree
• 5.
  N2 = 2N3 =4 N4 = N2+N3+1=7N1 = 1
• 6.
  Height of AVLTree Denote Nh the minimum number of nodes in an AVL tree of height h N1=1, N2 =2 (base) Nh= Nh-1 + Nh-2 +1 (recursive relation)  many operations (i.e. searching) on an AVL tree will take O(log N) time
• 7.
  Insertion in AVLTree Basically follows insertion strategy of binary search tree But may cause violation of AVL tree property Restore the destroyed balance condition if needed 6 7 6 8 Original AVL tree Insert 6 Property violated Restore AVL property
• 8.
  Some Observations After aninsertion, 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 Rebalance the tree at the deepest such node guarantees that the entire tree satisfies the AVL property 7 6 8 Rebalance node 7 guarantees the whole tree be AVL 6 Node 5,8,7 might have balance altered
• 9.
  Different Rebalancing Rotations The rebalancingrotations are classified as LL, LR, RR and RL as illustrated below, based on the position of the inserted node with reference to α. LL rotation: Inserted node is in the left subtree of the left subtree of node α RR rotation: Inserted node is in the right subtree of the right subtree of node α LR rotation: Inserted node is in the right subtree of the left subtree of node α RL rotation: Inserted node is in the left subtree of the right subtree of node α
• 10.
  Insertion Analysis Insert thenew key as a new leaf just as in ordinary binary search tree: O(logN) Then trace the path from the new leaf towards the root, for each node x encountered: O(logN) Check height difference: O(1) If satisfies AVL property, proceed to next node: O(1) If not, perform a rotation: O(1) The insertion stops when A single rotation is performed Or, we’ve checked all nodes in the path Time complexity for insertion O(logN) logN
• 11.
  Delete 5, Node10 is unbalanced Single Rotation 20 10 35 40155 25 18 453830 50 20 15 35 401810 25 453830 50  Basically follows deletion strategy of binary search tree  But may cause violation of AVL tree property  Restore the destroyed balance condition if needed
• 12.
  For deletion, afterrotation, we need to continue tracing upward to see if AVL-tree property is violated at other node. Different rotations are classified in R0, R1, R-1, L0, L1 And L-1 20 15 35 401810 25 453830 50 20 15 35 40 1810 25 4538 30 50 Continue to check parents Oops!! Node 20 is unbalanced!! Single Rotation
• 13.
  Different rotations • Anelement can be deleted from AVL tree which may change the BF of a node such that it results in unbalanced tree. • Some rotations will be applied on AVL tree to balance it. • R rotation is applied if the deleted node is in the right subtree of node A (A is the node with balance factor other than 0, 1 and -1). • L rotation is applied if the deleted node is in the left subtree of node A.
• 14.
  Different rotations cont… •Suppose we have deleted node X from the tree. • A is the closest ancestor node on the path from X to the root node, with a balance factor -2 or +2. • B is the desendent of node A on the opposite subtree of deleted node i.e. if the deleted node is on left side the B is the desendent on the right subtree of A or the root of right subtree of A.
• 15.
  R Rotation • RRotation is applied when the deleted node is in the right subtree of node A. • There are three different types of rotations based on the balanced factor of node B. • R0 Rotation: When the balance Factor of node B is 0. • Apply LL Rotation on node A. • R1 Rotation: When the balance Factor of node B is +1. • Apply LL Rotation on node A. • R-1 Rotation: When the balance Factor of node B is -1. • Apply LR Rotation(RR rotation on B and LL rotation on node A).
• 16.
  L Rotation • LRotation is applied when the deleted node is in the left subtree of node A. • There are three different types of rotations based on the balanced factor of node B. • L0 Rotation: When the balance Factor of node B is 0. • Apply RR Rotation on node A. • L-1 Rotation: When the balance Factor of node B is -1. • Apply RR Rotation on node A. • L1 Rotation: When the balance Factor of node B is +1. • Apply RL Rotation(LL rotation on B and RR rotation on node A).