Presentation is loading. Please wait.

Presentation is loading. Please wait.

CS3424 AVL Trees Red-Black Trees. Trees As stated before, trees are great ways of holding hierarchical data Insert, Search, Delete ~ O(lgN) But only if.

Published byEssence Branham Modified over 9 years ago

Similar presentations

Presentation on theme: "CS3424 AVL Trees Red-Black Trees. Trees As stated before, trees are great ways of holding hierarchical data Insert, Search, Delete ~ O(lgN) But only if."— Presentation transcript:

1 CS3424 AVL Trees Red-Black Trees

2 Trees As stated before, trees are great ways of holding hierarchical data Insert, Search, Delete ~ O(lgN) But only if they’re balanced! So let’s discuss how to assure balance

3 Rotations x y    y x    Left-Rotate(x) Right-Rotate(y)

4 AVL Trees (1962) Uses height property to maintain balance Height of left & right differ by at most 1 Uses rotations to restore balance after insertions and deletions

5 Balancing in AVL Trees Balance is the difference between the heights of the left and right subtrees –This should range between -1 and 1 Maintain this balance upon insert and delete Balance = H R - H L

6 Left-Left Imbalance X Y    -2 X Y    0 0 Right-Rotate(X)

7 Left-Right Imbalance X Y    +1 -2 X Y   +1 -2  =

8 Left-Right Imbalance (cont) X Y    +1 -2 X Y   +1 -2  = +1

9 Left-Right Imbalance (  left) X Y   +1 -2 

10 Left-Right Imbalance (  left) X Y   +1 -2  Left-Rotate(Y) X Y   -2  0

11 Left-Right Imbalance (  left) Right-Rotate(X) X Y   -2  0 X Y   0 0  +1

12 Left-Right Imbalance (  right) X Y   +1 -2  +1

13 Left-Right Imbalance (  right) X Y   +1 -2  +1 Left-Rotate(Y) X Y   -2 

14 Left-Right Imbalance (  right) Right-Rotate(X) X Y   -2  X Y   0 0 

15 Red-Black Trees (1972/1978) BST + notion of color (RED / BLACK) Every node is either red or black Root is black Every leaf (NIL) is black If node red, both children are black For each node, path to NIL children contains same number of black nodes (black-height)

16 Possibilities in Growing a Red- Black Tree

17 Painting a Tree to be Red-Black

18 Root is black

19 Painting a Tree to be Red-Black bh=1 Must be black

20 Painting a Tree to be Red-Black

21

22 Must be black

23 Painting a Tree to be Red-Black Must be red

24 Painting a Tree to be Red-Black bh = 2 or 3

25 Inserting & Deleting in Red- Black Trees Insert similar to binary search trees –“Search” left & right until finding NIL –But now, we need to worry about coloring –Requires possible rotations & “cleanup” Deleting similar to BSTs as well –Search left & right until finding item –If we removed a black node, “cleanup”

26 Red-Black Insert Insert node (z) z.Color = red –This can cause two possible problems: Root must be black (initial insert violates this) Two red nodes cannot be adjacent (this is violated if parent of z is red) –Cleanup

27 Red-Black Cleanup y = z’s “uncle” Three cases: –y is red –y is black and z is a right child –y is black and z is a left child Not mutually exclusive

28 Case 1 – z’s Uncle is red y.Color = black z.Parent.Color = black z.Parent.Parent.Color = red z = z.Parent.Parent Repeat cleanup

29 Case 1 Visualized 11 214 7115 58 4 z y y.Color = black z.Parent.Color = black z.Parent.Parent.Color = red z = z.Parent.Parent repeat cleanup New Node

30 Case 1 Visualized 11 214 7115 58 4 z y y.Color = black z.Parent.Color = black z.Parent.Parent.Color = red z = z.Parent.Parent repeat cleanup New Node

31 Case 1 Visualized 11 214 7115 58 4 z y y.Color = black z.Parent.Color = black z.Parent.Parent.Color = red z = z.Parent.Parent repeat cleanup New Node

32 Case 1 Visualized 11 214 7115 58 4 z y y.Color = black z.Parent.Color = black z.Parent.Parent.Color = red z = z.Parent.Parent repeat cleanup New Node

33 Case 1 Visualized 11 214 7115 58 4 z y y.Color = black z.Parent.Color = black z.Parent.Parent.Color = red z = z.Parent.Parent repeat cleanup New Node

34 Case 2 – z’s Uncle is black & z is a right child z = z.Parent Left-Rotate(T, z) Do Case 3 Note that Case 2 is a subset of Case 3

35 Case 2 Visualized 11 214 7115 58 4 z y z = z.Parent Left-Rotate(T,z) Do Case 3

36 Case 2 Visualized 11 214 7115 58 4 z y z = z.Parent Left-Rotate(T,z) Do Case 3

37 Case 2 Visualized 11 2 147 1 15 5 8 4 z y z = z.Parent Left-Rotate(T,z) Do Case 3

38 Case 3 – z’s Uncle is black & z is a left child z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

39 Case 3 Visualized 11 2 147 1 15 5 8 4 z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

40 Case 3 Visualized 11 2 147 1 15 5 8 4 z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

41 Case 3 Visualized 11 2 147 1 15 5 8 4 z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

42 Case 3 Visualized 112 14 7 1 15 5 8 4 z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

43 Analysis of Red-Black Cleanup A B    C D  Case 1 , , , , and  are all black-rooted and all have same black-height (symmetric L/R on A & B) z y A B    C D  New z

44 Analysis of Red-Black Cleanup A B    C  Case 2 , , , and  are all black-rooted and all have same black-height (  is black or this would be case 1) z y C  B A    z y Case 2 Case 3

45 Analysis of Red-Black Cleanup C  B A    z y Case 3 , , , and  are all black-rooted and all have same black-height B  AC   z y

46 Questions about Red-Black Insert What is the running time? Why are there only at most 2 rotations in the cleanup? How many times can the cleanup repeat (i.e. how many times can we have case 1 in a row)?

47 Red-Black Delete Delete as before But if you take away a black node: –Black height is not “balanced” –Can generate adjacent red nodes –Must perform cleanup 4 cases “Beyond the scope of this course…”

48 FIN

Download ppt "CS3424 AVL Trees Red-Black Trees. Trees As stated before, trees are great ways of holding hierarchical data Insert, Search, Delete ~ O(lgN) But only if."

Similar presentations

Chapter 13. Red-Black Trees

Chapter 13. Red-Black Trees
Introduction to Algorithms Red-Black Trees

Introduction to Algorithms Red-Black Trees
AVL Trees1 Part-F2 AVL Trees 6 3 8 4 v z. AVL Trees2 AVL Tree Definition (§ 9.2) AVL trees are balanced. An AVL Tree is a binary search tree such that.

AVL Trees1 Part-F2 AVL Trees v z. AVL Trees2 AVL Tree Definition (§ 9.2) AVL trees are balanced. An AVL Tree is a binary search tree such that.
Red-Black tree Recall binary search tree –Key values in the left subtree = the node value Operations:

Red-Black tree Recall binary search tree –Key values in the left subtree = the node value Operations:
AVL Trees CS II – Fall 2010 9/8/2010. Announcements HW#2 is posted – Uses AVL Trees, so you have to implement an AVL Tree class. Most of the code is provided.

AVL Trees CS II – Fall /8/2010. Announcements HW#2 is posted – Uses AVL Trees, so you have to implement an AVL Tree class. Most of the code is provided.
CS202 - Fundamental Structures of Computer Science II

CS202 - Fundamental Structures of Computer Science II
AA Trees another alternative to AVL trees. Balanced Binary Search Trees A Binary Search Tree (BST) of N nodes is balanced if height is in O(log N) A balanced.

AA Trees another alternative to AVL trees. Balanced Binary Search Trees A Binary Search Tree (BST) of N nodes is balanced if height is in O(log N) A balanced.
ITEC200 Week 11 Self-Balancing Search Trees. www.ics.mq.edu.au/ppdp 2 Learning Objectives Week 11 (ch 11) To understand the impact that balance has on.

ITEC200 Week 11 Self-Balancing Search Trees. 2 Learning Objectives Week 11 (ch 11) To understand the impact that balance has on.
Red-Black Trees CIS 606 Spring 2010. Red-black trees A variation of binary search trees. Balanced: height is O(lg n), where n is the number of nodes.

Red-Black Trees CIS 606 Spring Red-black trees A variation of binary search trees. Balanced: height is O(lg n), where n is the number of nodes.
Analysis of Algorithms CS 477/677 Instructor: Monica Nicolescu Lecture 11.

Analysis of Algorithms CS 477/677 Instructor: Monica Nicolescu Lecture 11.
CS 46101 Section 600 CS 56101 Section 002 Dr. Angela Guercio Spring 2010.

CS Section 600 CS Section 002 Dr. Angela Guercio Spring 2010.
Lecture 12: Balanced Binary Search Trees Shang-Hua Teng.

Lecture 12: Balanced Binary Search Trees Shang-Hua Teng.
Analysis of Algorithms CS 477/677 Red-Black Trees Instructor: George Bebis (Chapter 14)

Analysis of Algorithms CS 477/677 Red-Black Trees Instructor: George Bebis (Chapter 14)
1 /26 Red-black tree properties Every node in a red-black tree is either black or red Every null leaf is black No path from a leaf to a root can have two.

1 /26 Red-black tree properties Every node in a red-black tree is either black or red Every null leaf is black No path from a leaf to a root can have two.
1 /26 Red-black tree properties Every node in a red-black tree is either black or red Every null leaf is black No path from a leaf to a root can have two.

1 /26 Red-black tree properties Every node in a red-black tree is either black or red Every null leaf is black No path from a leaf to a root can have two.
1 Red-Black Trees. 2 Black-Height of the tree = 4.

1 Red-Black Trees. 2 Black-Height of the tree = 4.
1 2-3-4 Trees and Red-Black Trees Gordon College Prof. Brinton.

Trees and Red-Black Trees Gordon College Prof. Brinton.
Self-Balancing Search Trees Chapter 11. Chapter 11: Self-Balancing Search Trees2 Chapter Objectives To understand the impact that balance has on the performance.

Self-Balancing Search Trees Chapter 11. Chapter 11: Self-Balancing Search Trees2 Chapter Objectives To understand the impact that balance has on the performance.
Fall 2007CS 2251 Self-Balancing Search Trees Chapter 9.

Fall 2007CS 2251 Self-Balancing Search Trees Chapter 9.
Self-Balancing Search Trees Chapter 11. Chapter Objectives  To understand the impact that balance has on the performance of binary search trees  To.

Self-Balancing Search Trees Chapter 11. Chapter Objectives  To understand the impact that balance has on the performance of binary search trees  To.

Similar presentations

Ads by Google