1. Computer Science 2-3-4 Red-Black CS 330: Algorithms 2-3-4 and Red-Black Trees Gene Itkis

2. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis2 2-3-4 trees  2-3-4  2-3-4 nodes:  Red-black  Red-black implementation: x y z >y<z>y<z >z>z >x<y>x<y <x<x >y>y >x<y>x<y <x<x xyxy >x>x<x<x x >x>x<x<x x >y>y >x<y>x<y <x<x x y y x or >y<z>y<z >z>z >x<y>x<y <x<x y z x

3. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis3 Red-Black trees  Properties:  Every node is either red or black  Root: black  Leaf (nil): black  Children of red node are black  Any root-leaf path has same # of black nodes  Black depth of node v  = “# of black nodes on the path from root to v  same for all leaves  Black2-3-4  Black node = “root” of a 2-3-4 node  Black2-3-4  Black depth = depth in 2-3-4 tree

4. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis4 2-3-4Red-Black 2-3-4 & Red-Black trees  2-3-4  2-3-4 trees  Perfectly balanced  height ≤ lg n  Red-Black  Red-Black trees  2-3-4  2-3-4 node = Red-Black  =Red-Black subtree of height ≤ 2subtree  Red-Blackheight ≤ 2 lg n  Red-Black tree height ≤ 2 lg n  Search (both 2-3-4Red-Black: O(lg n)  Search (both 2-3-4 & Red-Black): O(lg n)

5. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis5 2-3-4 trees  2-3-4  2-3-4 nodes:  Red-black  Red-black implementation: x y z >y<z>y<z >z>z >x<y>x<y <x<x >y>y >x<y>x<y <x<x xyxy >x>x<x<x x >x>x<x<x x >y>y >x<y>x<y <x<x x y y x or >y<z>y<z >z>z >x<y>x<y <x<x y z x

6. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis6 2-3-4 2-3-4 tree example52 7 8 6 9 5 225225 7 2 5 7 7 2 5 878878 2 5 6 6 7 8 2 5 2 5 8 6 7 5757 2 8 6 9

7. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis7 B Zoom-in: one node  Simple inserts:  Case 0  No fixing needed  …even inside tree  More complex:  Case 1:  Split (3,4,5)  4 will try to join higher  6 can now join as case 0  …same inside tree 5 6+66+6 5 6 55 6 4 4 4 4 6 A CDB A CDB A 5 4 6+66+6 4 5 3 4 6 3 3 4 5 3 6 5 B A CB A C 6 DEDE

8. Computer Science CS-330: Algorithms, Fall 2004Gene Itkis8 Zoom-in: one node  Case 0:  Case 3:  Rotate 4-5…  To get case 0 above  Case 2:  Rotate 6-5  To get case 3 above 5 6+66+6 5 6 5 5 6 4 4 4 4 6 5 4 4 5 6+66+6 5 6 4 5 6 4 5 4 5 6 4 4 6 5+55+5 5 6 4 5 6 4 6 6