CS350: Data Structures AVL Trees

Transcription

1 S35: Data Structures VL Trees James Moscola Department of Engineering & omputer Science York ollege of Pennsylvania S35: Data Structures James Moscola

2 Balanced Search Trees Binary search trees are not guaranteed to be balanced given random insertions and deletions - Inserting a sorted lists of elements into a BST produces the worst case -- O(N) - Performance of an unbalanced tree can degrade as more elements are inserted Balanced search tree operations, such as insert, insure that the a tree always remains balanced - n operation is not complete until it returns the tree to a balanced state Balanced tree Unbalanced tree S35: Data Structures 2

3 VL Trees type of balanced binary search tree Named for its discoverers -- delson, Velskii, and Landis Definition: n VL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. S35: Data Structures 3

4 VL Trees 2 3 The height of each node is listed in blue n VL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. S35: Data Structures 4

5 VL Trees 2 3 The height of each node is listed in blue height differs by at most n VL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. S35: Data Structures 5

6 VL Trees 2 3 The height of each node is listed in blue height differs by at most n VL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. S35: Data Structures 6

7 VL Trees The binary tree after inserting node 2 What are the node heights S35: Data Structures 7

8 VL Trees The binary tree after inserting node 4 2 What are the node heights S35: Data Structures 8

9 VL Trees 4 2 height differs by more than therefore, node 8 is invalid S35: Data Structures 9

10 VL Trees 4 2 height differs by more than therefore, node 8 is unbalanced height differs by more than therefore, node 2 is unbalanced 2 6 S35: Data Structures

11 VL Trees When a node in the tree no longer satisfies the invariant required to be an VL tree the tree must be rebalanced around that node There are four ways in which an insertion into a tree may cause an imbalance to occur at some node X: () n insertion in the left subtree of the left child of X (2) n insertion in the right subtree of the left child of X (3) n insertion in the left subtree of the right child of X (4) n insertion in the right subtree of the right child of X symmetric S35: Data Structures

12 reating an Imbalance -- ase # () n insertion in the left subtree of the left child of X N2 N2 2 N N B B root of subtree increases height as does N and N2 Node N and root of differ by more than therefore N2 violates VL rules S35: Data Structures 2

13 Fixing the Imbalance -- ase # Perform a single right rotation N becomes new root node N2 2 N N2 now right child of N N N2 B B B subtree now left child of N2 S35: Data Structures 3

14 reating an Imbalance -- ase #4 (symmetric with ) (4) n insertion in the right subtree of the right child of X N2 N N2 N N B root of subtree increases height as does N and N2 Node N and root of differ by more than therefore N2 violates VL rules B S35: Data Structures 4

15 Fixing the Imbalance -- ase #4 (symmetric with ) Perform a single left rotation N2 N2 now left child of N N N becomes new root node N N2 B B B subtree now right child of N2 S35: Data Structures 5

16 reating an Imbalance -- ase #2 (2) n insertion in the right subtree of the left child of X N3 2 N3 N N B Node N and root of differ by more than therefore N3 violates VL rules B root of B subtree increases height as does N and N3 S35: Data Structures 6

17 Fixing the Imbalance -- ase #2 We know the node N2 exists because a node was just inserted into the B subtree N3 2 2 N3 N N N2 B B B2 S35: Data Structures 7

18 Fixing the Imbalance -- ase #2 Perform a left-right double rotation N3 2 Step # - Perform a single left rotation between N and N2 N3 2 tree is still unbalanced N N2 B N2 B2 N B B2 S35: Data Structures 8

19 Fixing the Imbalance -- ase #2 Perform a left-right double rotation N3 2 Step #2 - Perform a single right rotation between N2 and N3 N2 N2 N N3 N B B2 B B2 S35: Data Structures 9

20 Fixing the Imbalance -- ase #2 nother look at the left-right double rotation without the intermediate steps N3 N2 N N N3 N2 B B2 B B2 S35: Data Structures 2

21 Fixing the Imbalance -- ase #3 (symmetric with 2) Here is a look at the right-left double rotation without the intermediate steps N3 N2 N N3 N N2 B B2 B B2 S35: Data Structures 2

22 VL Tree Insertion Example S35: Data Structures 22

23 VL Tree Insertion Example Inserting node - First, recursively search for the location which to insert the node - Next, insert the node - Finally, unwind the recursion update node heights along the way. Rebalance tree where necessary To update node s height, check its children S35: Data Structures 23

24 VL Tree Insertion Example node is a leaf node so its height is 2 6 S35: Data Structures 24

25 VL Tree Insertion Example node 2 - add to the height of its highest child S35: Data Structures 25

26 VL Tree Insertion Example 2 node 4 - add to the height of its highest child S35: Data Structures 26

27 VL Tree Insertion Example node 8 violates VL properties, so rebalance 2 height differs by more than S35: Data Structures 27

28 VL Tree Insertion Example ase #: Must perform single right rotation between node 8 and node S35: Data Structures 28

29 VL Tree Insertion Example fter rotation, update height of node 8 Node 4 is balanced, so continue unwinding recursion S35: Data Structures 29

30 VL Tree Insertion Example node 2 - add to the height of its highest child S35: Data Structures 3

31 Other Operations: find / remove The find operation is the same as the unbalanced binary search tree The remove operation works similarly to the remove operation from the unbalanced binary search tree with a few modifications - When a node is removed, the heights of its ancestors may need to be updated as the recursion is unwound -- fix imbalances as they are encountered just like with insertion S35: Data Structures 3

32 nalysis of VL Tree Operations Time complexity of VL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) remove O(log N) O(log N) S35: Data Structures 32