Data Structures and Algorithms

Transcription

1 Data Structures and Algorithms Spring

2 Outline BST Trees (contd.) 1 BST Trees (contd.)

3 Outline BST Trees (contd.) 1 BST Trees (contd.)

4 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder

5 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder

6 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder

7 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: M D N B Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Q

8 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: M AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder D

9 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder

10 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder

11 The bad news about BSTs... Problem with BSTs is that there is nothing to stop them going badly skewed An AVL (Adelson-Velskii and Landis) tree is a BST with a height balance condition (There exist other balancing strategies based on the weight of a tree) Cannot just insist that the root be balanced: Also, cannot insist that every node have an equal height left and right subtree: AVL requirement is that the height of every left and right subtree can differ by at most 1 Height information is kept in tree_node object Insertions and deletions now become harder

12 AVL Tree Example BST Trees (contd.) M M M D N D N D N B G B G B G O H H AVL Tree Not an AVL Tree AVL Tree In practise an AVL tree on n nodes has height of about O(log(n + 1)) Compare to best BST (CBBST) which has height k = log(n + 1) 1 The worst the height can be is 1.44 log n found by reasoning on what is the minimum number of nodes that a tree of height, h, can have; we will return to this problem later

13 : Insertions When we insert a node in to an AVL tree we need to ensure that the tree remains balanced after the insertion Consider inserting values 1 to 7 in to an empty AVL Inserting values 1 and 2 are easy, but 3... Inserting 6:

14 : Insertions When we insert a node in to an AVL tree we need to ensure that the tree remains balanced after the insertion Consider inserting values 1 to 7 in to an empty AVL Inserting values 1 and 2 are easy, but 3... Inserting 6:

15 : Insertions When we insert a node in to an AVL tree we need to ensure that the tree remains balanced after the insertion Consider inserting values 1 to 7 in to an empty AVL Inserting values 1 and 2 are easy, but Tree is unbalanced at node 1 Inserting 6: Rotate tree about node 2 to accommodate node 3

16 : Insertions When we insert a node in to an AVL tree we need to ensure that the tree remains balanced after the insertion Consider inserting values 1 to 7 in to an empty AVL Inserting values 1 and 2 are easy, but 3... Inserting 6: Unbalanced at node 2 6 Rotate tree about node 4 to accommodate node 6

17 : Single Rotations k 2 k 1 k 1 Z X k 2 X Y Y Z If the height of k 2 s left subtree is two more than the right subtree then there is a left imbalance and we perform a left rotation, giving the picture on the right If the height of k 1 s right subtree is two more than the left subtree then there is a right imbalance and we perform a right rotation, giving the picture on the left

18 : Insertions (contd.) After inserting 7, suppose we want to insert 15, 14,..., 8 Inserting 14: A single rotation will not repair the balancedness of the tree so we need to perform a deeper twist This is called a double rotation There are two types of double rotation: right-left rotations and left-right rotations (think in terms of imbalanced side) A double rotation is the composition of two single rotations

19 : Insertions (contd.) After inserting 7, suppose we want to insert 15, 14,..., 8 Inserting 14: Single Rotation A single rotation will not repair the balancedness of the tree so we need to perform a deeper twist This is called a double rotation There are two types of double rotation: right-left rotations and left-right rotations (think in terms of imbalanced side) A double rotation is the composition of two single rotations 14

20 : Insertions (contd.) After inserting 7, suppose we want to insert 15, 14,..., 8 Inserting 14: A single rotation will not repair the balancedness of the tree so we need to perform a deeper twist This is called a double rotation There are two types of double rotation: right-left rotations and left-right rotations (think in terms of imbalanced side) A double rotation is the composition of two single rotations

21 : Insertions (contd.) After inserting 7, suppose we want to insert 15, 14,..., 8 Inserting 14: A single rotation will not repair the balancedness of the tree so we need to perform a deeper twist This is called a double rotation There are two types of double rotation: right-left rotations and left-right rotations (think in terms of imbalanced side) A double rotation is the composition of two single rotations

22 : Rotations (contd.) A left-right rotation is the composition of a right rotation on the subtree rooted at k 1 followed by a left rotation on the subtree rooted at k 3 (upper case on picture over) A right-left rotation is the composition of a left rotation on the subtree rooted at k 3 followed by a right rotation on the subtree rooted at k 1 (lower case on picture over)

23 : Rotations (contd.) A left-right rotation is the composition of a right rotation on the subtree rooted at k 1 followed by a left rotation on the subtree rooted at k 3 (upper case on picture over) A right-left rotation is the composition of a left rotation on the subtree rooted at k 3 followed by a right rotation on the subtree rooted at k 1 (lower case on picture over)

24 : Rotations (contd.) k 3 W k 1 k 2 Z Left Right double rotation X Y k 2 k 1 k 3 k 1 W X Y Z W k 2 k 3 Z Right Left double rotation X Y

25 AVL Tree: Deletions BST Trees (contd.) Just as with BSTs, deletions are similar to, but more tricky than, insertions The actual deletion is simply that described earlier for the BST However, shifting the right subtree s leftmost descendant to maintain the inorderedness may cause an imbalance Restoring the balance (through rotations) may cause imbalance further up the tree While it will never take more than one single or double rotation to rebalance a tree after an insertion, we may need h 2 rotations to rebalance a tree of height h after a deletion