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Red Black Trees

Red Black Trees. Colored Nodes Definition Binary search tree. Each node is colored red or black. Root and all external nodes are black. No root-to-external-node path has two consecutive red nodes. All root-to-external-node paths have the same number of black nodes. Red Black Trees.

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Red Black Trees

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  1. Red Black Trees Colored Nodes Definition • Binary search tree. • Each node is colored red or black. • Root and all external nodes are black. • No root-to-external-node path has two consecutive red nodes. • All root-to-external-node paths have the same number of black nodes

  2. Red Black Trees Colored Edges Definition • Binary search tree. • Child pointers are colored red or black. • Pointer to an external node is black. • No root to external node path has two consecutive red pointers. • Every root to external node path has the same number of black pointers.

  3. 10 7 40 45 3 8 30 60 35 1 20 5 25 Example Red-Black Tree

  4. Properties • The height of a red black tree that has n (internal) nodes is between log2(n+1) and 2log2(n+1).

  5. Properties • Start with a red black tree whose height is h; collapse all red nodes into their parent black nodes to get a tree whose node-degrees are between 2 and 4, height is >= h/2,and all external nodes are at the same level.

  6. 10 7 40 45 3 8 30 60 35 1 20 5 25 Properties

  7. Properties • Let h’ ≥ h/2 be the height of the collapsed tree. • Internal nodes of collapsed tree have degree between 2 and 4. • Number of internal nodes in collapsed tree ≥ 2h’-1. • So, n≥ 2h’-1 • So, h ≤ 2 log2 (n + 1)

  8. Properties • At most 1 rotation and O(log n) color flips per insert/delete. • Priority search trees. • Two keys per element. • Search tree on one key, priority queue on other. • Color flip doesn’t disturb priority queue property. • Rotation disturbs priority queue property. • O(log n) fix time per rotation => O(log2n) overall time for AVL.

  9. Properties • Every AVL tree is a red-black tree. • A binary tree is red-black iff for every node x, the length of the longest path from x to an external node in its subtree is at most twice that of the shortest. • O(1) amortized complexity to restructure following an insert/delete. • C++ STL implementation • java.util.TreeMap => red-black tree

  10. Insert • New pair is placed in a new node, which is inserted into the red-black tree. • New node color options. • Black node => one root-to-external-node path has an extra black node (black pointer). • Hard to remedy. • Red node => one root-to-external-node path may have two consecutive red nodes (pointers). • May be remedied by color flips and/or a rotation.

  11. gp pp d p c a b Classification Of 2 Red Nodes/Pointers LLb • XYz • X => relationship between gp and pp. • pp left child of gp => X = L. • Y => relationship between pp and p. • p left child of pp => Y = L. • z = b (black) if d = null or a black node. • z = r (red) if d is a red node.

  12. gp pp d gp p c pp d p a b c a b XYr • Color flip. • Move p, pp, and gp up two levels. • Continue rebalancing.

  13. gp y z pp x z y d p x a b c d c a b LLb • Rotate. • Done! • Same as LL rotation of AVL tree.

  14. gp y z pp x z x d a b c d a p y b c LRb • Rotate. • Done! • Same as LR rotation of AVL tree. • RRb and RLb are symmetric.

  15. Delete • Delete as for unbalanced binary search tree. • If red node deleted, no rebalancing needed. • If black node deleted, a subtree becomes one black pointer (node) deficient.

  16. 10 7 40 45 3 8 30 60 35 1 20 5 25 Delete A Black Leaf • Delete 8.

  17. 10 7 40 45 3 30 60 35 1 20 5 25 py Delete A Black Leaf y • y is root of deficient subtree. • py is parent of y.

  18. 10 7 40 45 3 8 30 60 35 1 20 5 25 py Delete A Black Degree 1 Node y • Delete 45. • y is root of deficient subtree.

  19. 10 7 40 45 3 8 30 60 35 1 20 5 25 Delete A Black Degree 2 Node • Not possible, degree 2 nodes are never deleted.

  20. 10 7 40 45 3 8 30 py 60 35 1 20 5 25 y Rebalancing Strategy • If y is a red node, make it black.

  21. 10 py 7 40 45 3 8 30 y 60 35 1 20 5 25 Rebalancing Strategy • Now, no subtree is deficient. Done!

  22. y 10 7 40 45 3 8 30 60 35 1 20 5 25 Rebalancing Strategy • y is a black root (there is no py). • Entire tree is deficient. Done!

  23. py v y a b Rebalancing Strategy • y is black but not the root (there is a py). • Xcn • y is right child of py=> X = R. • Pointer to v is black => c = b. • v has 1 red child => n = 1.

  24. py py v y v y a b a b Rb0 (case 1, py is black) • Color change. • Now, py is root of deficient subtree. • Continue!

  25. py py v y v y a b a b Rb0 (case 2, py is red) • Color change. • Deficiency eliminated. • Done!

  26. v py py v a y b y a b Rb1 (case 1) • LL rotation. • Deficiency eliminated. • Done!

  27. w py py v v y c y a b w a b c Rb1 (case 2) • LR rotation. • Deficiency eliminated. • Done!

  28. w py py v v y c y a b w a b c Rb2 • LR rotation. • Deficiency eliminated. • Done!

  29. py v y w a b c Rr(n) • n = # of red children of v’s right child w. Rr(2)

  30. v py py v a y b y a b Rr(0) • LL rotation. • Done!

  31. w py py v v y c y a b w a c b Rr(1) (case 1) • LR rotation. • Deficiency eliminated. • Done!

  32. x py py v v y d y a w w a x b b c c d Rr(1) (case 2) • Rotation. • Deficiency eliminated. • Done!

  33. x py py v v y d y a w w a x b b c c d Rr(2) • Rotation. • Deficiency eliminated. • Done!

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