1. IKI 10100: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1 Fasilkom UI Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red Black Tree

2. 2 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black Trees Motivation Definition Operation Outline

3. 3 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Binary Search Trees should be balanced. AVL Trees need 2 passes: top-down insertion/deletion and bottom-up rebalancing  Need recursive implementation Red-Black Trees need 1 pass: top-down rebalancing and insertion/deletion  Can be implemented iteratively, faster Red-Black Trees have slightly weaker balance restrictions  Less effort to maintain  In practice, worst case is similar to AVL Trees Motivation

4. 4 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black Trees: Definition Rules of Red-Black Trees: 1. Every node is colored either red or black 2. The root is black 3. If a node is red, its children must be black consecutive red nodes are disallowed 4. Every path from a node to a null reference must contain the same number of black nodes Convention: null nodes are black

5. 5 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 550 10 70 8560 4055 809065 20 Red-Black Trees The insertion sequence is 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55

6. 6 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black Trees: Properties Each path must contain the same number of black nodes. (Rule #4) Consecutive red nodes are not allowed. (Rule #3) The longest path is at most twice the length of the shortest path

7. 7 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black Trees: Properties B = total black nodes from root to leaf N = total all nodes H = height All operations guaranteed logarithmic!

8. 8 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Insertion A new node must be colored red Why? A new item is always inserted as a leaf in the tree If we color a new item black, then the number of black nodes from root would be different (violate property #4) If the parent is black, no problem. If the parent is red, we create two consecutive red nodes (violate property #3) Thus, we have to do some rotating/recolouring… Remember convention: null nodes are black

9. 9 Ruli Manurung (Fasilkom UI)IKI10100: Lecture AB G X S P C DE AB G X S P C DE Single Rotation X: new node P: parent S: sibling G: Grandparent Case after insertion: Consecutive red (P & X) Sibling of parent (S) is black X is “outer node” (left-left or right-right)

10. 10 Ruli Manurung (Fasilkom UI)IKI10100: Lecture BC G X S P A DE AB G P S X C DE Double Rotation X: new node P: parent S: sibling G: Grandparent Case after insertion: Consecutive red (P & X) Sibling of parent (S) is black X is “inner node” (left-right or left)

11. 11 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Single Rotation (bottom-up) Case after insertion: Consecutive red Sibling of parent is red Outer node (left-left or right-right) AB G X SP C DE AB G X S P C DE But what if P’s parent is red?  We have to keep going up the tree all the way to the root 

12. 12 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Top-Down Insertion The solution: prevent S from ever being red! Starting from the root (searching for insertion point) Never allow 2 red siblings If we see a node X with 2 red children, do a colour flip. X C2C1 X C2

13. 13 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Color Flip Maintains property #4 Possible violation of #3: if X’s parent is red! Do single or double rotation X’s parent’s sibling can never be red! Set the root to black (to maintain property #2) X C2C1 X C2

14. 14 Ruli Manurung (Fasilkom UI)IKI10100: Lecture AB G X SP C DE Color Flip (2) AB G X SP C DE If we do the colour flipping on the way down to the insertion point, we will never reach a condition where P & S are red!

15. 15 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 5 10 70 85 60 50 55 8090 65 20 40 Example: Insert 18 18

16. 16 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 5 10 70 85 60 50 55 8090 65 20 40 Example: Insert 2 18 2

17. 17 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 2 5 70 85 60 50 55 8090 65 20 40 Example: Insert 2 18 10

18. 18 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 2 5 70 85 60 50 55 8090 65 20 40 Example: Insert 45 (Illustration) 18 10 45

19. 19 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 2 5 70 85 60 50 55 8090 65 20 40 Example: Insert 45 (Top-Down Color Flip) 18 10 45 Color-flip!

20. 20 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 2 5 70 85 60 50 55 8090 65 20 40 Example: Insert 45 (Top-Down Color Flip) 18 10 45 Color-flip!

21. 21 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 2 5 70 85 60 50 55 8090 65 20 40 Example: Insert 45 (Single Rotate) 18 10 45

22. 22 Ruli Manurung (Fasilkom UI)IKI10100: Lecture 30 15 2 5 70 85 60 50 55 8090 65 20 40 Example: Insert 45 (Single Rotate) 18 10 45

23. 23 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black Tree Insertion: Exercise The insertion sequence is 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55

24. 24 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black Tree: Deletion Deletion in BST: only leaf nodes or nodes with one child are really deleted (Why?) If the deleted node is red: no problem (all properties maintained). Leaf nodes: Single child nodes:

25. 25 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Top-Down Deletion If node to be deleted is black  violate property #4 Always ensure that the node to be deleted is red. Top-down traversal from root (looking for node to be deleted): X: visited node P: parent S: sibling B P S X A CD Idea: make sure that X is red!

26. 26 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Possible cases P is red (inductive invariant) X and S are black (result of property #3) 2 cases: 1. Both X’s children (A & B) are black 2. X has at least one red child (A, B, or both) B P S X A CD

27. 27 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Case 1: Both X’s children are black Depends on children of S (C & D): Both C & D are black: simply colour-flip: B P S X A C D B P S X A C D

28. 28 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Case 1: Both X’s children are black Outer child of S (C) is Red: do a single rotation and recolour B P S X A C D C S D P B X A

29. 29 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Case 1: Both X’s children are black Inner child of S (C) is Red: do a double rotation and recolour B P S X A D C C S P B X A D

30. 30 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Case 2: One/Both of X’s children is red Recurse down to X’s child If we land on a red node, fine. If we land on a black node, rotate sibling and parent: B P S X A C C S P B X A D D

31. 31 Ruli Manurung (Fasilkom UI)IKI10100: Lecture Red-Black trees use color as balancing information instead of height in AVL trees. An insertion may cause a local perturbation (two consecutive red nodes) The pertubation is either resolved locally (rotations), or propagated to a higher level in the tree by recoloring (color flip) O(1) for a rotation or color flip At most one restructuring per insertion. O(log n) color flips Total time: O(log n) Summary

32. 32 Ruli Manurung (Fasilkom UI)IKI10100: Lecture http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/resources/animation/data- structure/redblack/redblack.html (mirror from http://www.ece.uc.edu/~franco/C321/html/ RedBlack/redblack.html ) http://telaga.cs.ui.ac.id/WebKuliah/IKI101 00/resources/animation/data- structure/redblack/redblack.html Chapter 19.5 Further Reading