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.

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    X Y    =

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

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

10 Left-Right Imbalance (  left) X Y    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

13 Left-Right Imbalance (  right) X Y    +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 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 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 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 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 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 z y z = z.Parent Left-Rotate(T,z) Do Case 3

36 Case 2 Visualized z y z = z.Parent Left-Rotate(T,z) Do Case 3

37 Case 2 Visualized 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 z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

40 Case 3 Visualized z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

41 Case 3 Visualized z y z.Parent.Color = black z.Parent.Parent.Color = red Right-Rotate(T, z.Parent.Parent)

42 Case 3 Visualized 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


Ads by Google