Red-black-tree presentation in Algorithm
- 1. Balanced Search Trees R-B Trees Subject- Design and Analysis of Algorithms Submitted by- Prabhat Mishra Abhiraj Chaudhry
- 2. RED-BLACK TREES Definition: a binary search tree with nodes colored red and black such that: the paths from the root to any leaf have the same number of black nodes, there are no two consecutive red nodes, If a node is red, then both of its children are black the root is black.
- 3. A red-black tree of height for h even and h odd The total number of leaves is of the form: 1 + 2i1 + 2i2 + 2i3 +· · ·+2ih , So for h even the number of leaves is: 1 + 2(20 + 21 + 22 +· · ·+2(h/2)−1) = 2(h/2)+1 − 1, and for h odd it is: 1 + 2(20 + 21 + 22 +· · ·+2((h−1)/2)−1 ) + 2(h−1)/2 = 2(h−1)/2 − 1 So the worst-case height of a red-black tree is really 2 log n − O(1).
- 4. Definition: The black-height of a node, x, in a red-black tree is the number of black nodes on any path to a leaf, not counting x. The height of red-black tree Black-Height of the root = 2
- 5. Black-Height of the root = 3
- 6. Height-balanced trees We have to maintain the following balancedness property - Each path from the root to a leaf contains the same number of black nodes 7 3 18 10 22 26 11 8 bh =2 bh =2 bh =1 bh =1 bh =1 bh =1 bh =1
- 7. • Deleting a node from a red-black tree is a bit more complicated than inserting a node. - If the node is red? Not a problem – no RB properties violated - If the node is black? deleting it will change the black-height along some path
- 8. * We have some cases for deletion P S U V Case A: - V’s sibling, S, is Red Rotate S around P and recolor S & P delete
- 9. P S V P S V Rotate S around P P V S Recolor S & P
- 10. P S U V Case B: - V’s sibling, S, is black and has two black children. Recolor S to be Red delete Red or Black and don’t care
- 11. P S V P S V Recolor S to be Red
- 12. P S U V Case C: - S is black S’s RIGHT child is RED (Left child either color) Rotate S around P Swap colors of S and P, and color S’s Right child Black delete
- 13. P S V P S V Rotate S around P P S V Recolor: Swap colors of S and P, and color S’s Right child Black
- 14. P S U V delete Case D: - S is Black, S’s right child is Black and S’s left child is Red i) Rotate S’s left child around S ii) Swap color of S and S’s left child
- 15. P S V P S V Rotate S’s left child around S P S V Recolor: Swap color of S and S’s left child
- 16. Analysis of deletion - A red-black tree has O(log n) height - Search for deletion location takes O(log n) time - The swaping and deletion is O(1). - Each rotation or recoloring is O(1). - Thus, the deletion in a red-black tree takes O(log n) time