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Red-Black Trees Binary Search Trees with O(log n) Worst-Case Time per Operation The Red-Black Tree Properties Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. nil nil nil nil nil

  1. Red-Black Trees Binary Search Trees with O(log n) Worst-Case Time per Operation

  2. The Red-Black Tree Properties Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. nil nil nil nil nil Both children of a red node are black. nil nil nil nil nil Every root-to-leaf path contains the same nil nil number of black nodes. The black-height of a node x is the number of black nodes on a path from, but not including, x to a leaf. The black-height of T is the black-height of its root.

  3. The Height of a Red-Black Tree with n Nodes Lemma: A red-black tree of black-height h contains at least 2 h – 1 nodes. Lemma: A red-black tree of black-height h has height at most 2 h. nil nil nil nil nil Corollary: A red-black tree with n nodes has height at most 2 log ( n + 1) . nil nil nil nil nil nil nil Corollary: Operations Tree-Minimum, Tree-Maximum, Tree-Predecessor, Tree- Successor, and Find take O( log n) time on a red-black tree of size n.

  4. Rotations—The Key to Height Reduction Right rotation d b b e a d a c c e C A A B B C Left rotation

  5. A Rotation Takes Constant Time Right rotation d b b e a d a c c e C A A B B C d b e a c

  6. A Rotation Takes Constant Time Right rotation d b b e a d a c c e C A A B B C d b e a c

  7. A Rotation Takes Constant Time Right rotation d b b e a d a c c e C A A B B C d b e a c

  8. A Rotation Takes Constant Time Right rotation d b b e a d a c c e C A A B B C d b e a c

  9. A Rotation Takes Constant Time Right rotation d b b e a d a c c e C A A B B C b a d c e

  10. Insertion into a Red-Black Tree 17 19 7 13 18 5 37 nil nil nil nil nil 1 49 25 nil nil nil nil nil 33 nil nil Element to insert: 21

  11. Insertion into a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 18 The leaves are black. 5 37 nil nil nil nil nil Both children of a red node are black. 1 49 25 nil nil nil nil Every root-to-leaf path contains the 21 33 same number of black nodes. nil nil nil nil Element to insert: 21

  12. Insertion into a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 18 The leaves are black. 5 37 nil nil nil nil nil Both children of a red node are black. 1 49 25 nil nil nil nil nil Every root-to-leaf path contains the 33 same number of black nodes. nil nil Element to insert: 3

  13. Insertion into a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 18 The leaves are black. 5 37 No red-red property 8 nil nil nil nil nil Both children of a red node are black. 1 49 25 nil nil nil nil Every root-to-leaf path contains the 4 33 3 same number of black nodes. nil nil nil nil Goal: Restore no-red-red condition while maintaining the other four. Element to insert: 3 Tools: Recoloring of nodes; rotations

  14. Insertion into a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 18 The leaves are black. 5 37 No red-red property 8 nil nil nil nil nil Both children of a red node are black. 1 49 25 nil nil nil nil Every root-to-leaf path contains the 4 33 3 same number of black nodes. nil nil nil nil Goal: Restore no-red-red condition while maintaining the other four. Element to insert: 3 Tools: Recoloring of nodes; rotations

  15. Restoring the No-Red-Red Property Invariant: There is exactly one red node x in the tree whose parent may be red. Strategy: ■ Fix the violation of no-red-red property at x ■ This may violate the condition at some ancestor ■ Continue fixing the property at the ancestor (the ancestor becomes x ) Observations: ■ Initially, x has a red parent; hence, x is not the root. ■ If parent[ x ] is red, it is not the root either; parent[parent[ x ]] ? that is, parent[parent[ x ]] exists. ■ If parent[ x ] is black, we are done. parent[ x ] x

  16. Restoring the No-Red-Red Property Invariant: There is exactly one red node x in the tree whose parent may be red. Strategy: ■ Fix the violation of no-red-red property at x ■ This may violate the condition at some ancestor ■ Continue fixing the property at the ancestor (the ancestor becomes x ) Observations: ■ Initially, x has a red parent; hence, x is not the root. ■ If parent[ x ] is red, it is not the root either; parent[parent[ x ]] ? that is, parent[parent[ x ]] exists. ■ If parent[ x ] is black, we are done. parent[ x ] Assumptions: x ■ Neither x nor parent[x] is the root ■ parent[x] is the left child of parent[parent[x]]

  17. Restoring the No-Red-Red Property Invariant: There is exactly one red node x in the tree whose parent may be red. Strategy: ■ Fix the violation of no-red-red property at x ■ This may violate the condition at some ancestor ■ Continue fixing the property at the ancestor (the ancestor becomes x ) Observations: ■ Initially, x has a red parent; hence, x is not the root. ■ If parent[ x ] is red, it is not the root either; parent[parent[ x ]] that is, parent[parent[ x ]] exists. ■ If parent[ x ] is black, we are done. parent[ x ] Assumptions: x ■ Neither x nor parent[x] is the root ■ parent[x] is the left child of parent[parent[x]]

  18. Restoring the No-Red-Red Property Invariant: There is exactly one red node x in the tree whose parent may be red. Strategy: ■ Fix the violation of no-red-red property at x ■ This may violate the condition at some ancestor ■ Continue fixing the property at the ancestor (the ancestor becomes x ) Observations: ■ Initially, x has a red parent; hence, x is not the root. ■ If parent[ x ] is red, it is not the root either; parent[parent[ x ]] that is, parent[parent[ x ]] exists. ■ If parent[ x ] is black, we are done. parent[ x ] Assumptions: x ■ Neither x nor parent[x] is the root ■ parent[x] is the left child of parent[parent[x]]

  19. Restoring the No-Red-Red Property 3 cases: ■ Both parent[ x ] and its parent[parent[ x ]] x sibling are red Recoloring ■ parent[ x ] is red, its sibling parent[ x ] is black, and x is the left child of its parent x ■ As Case 2; but x is the Case 1 right child of its parent

  20. Restoring the No-Red-Red Property 3 cases: ■ Both parent[ x ] and its parent[parent[ x ]] x sibling are red Recoloring ■ parent[ x ] is red, its sibling parent[ x ] is black, and x is the left child of its parent x ■ As Case 2; but x is the Case 1 right child of its parent parent[parent[ x ]] Recoloring Right-rotation parent[ x ] x Case 2

  21. Restoring the No-Red-Red Property 3 cases: ■ Both parent[ x ] and its parent[parent[ x ]] x sibling are red Recoloring ■ parent[ x ] is red, its sibling parent[ x ] is black, and x is the left child of its parent x ■ As Case 2; but x is the Case 1 right child of its parent parent[parent[ x ]] Recoloring + Right-rotation parent[ x ] x Case 2

  22. Restoring the No-Red-Red Property 3 cases: ■ Both parent[ x ] and its parent[parent[ x ]] x sibling are red Recoloring ■ parent[ x ] is red, its sibling parent[ x ] is black, and x is the left child of its parent x ■ As Case 2; but x is the Case 1 right child of its parent parent[parent[ x ]] Recoloring + Left-rotation Right-rotation parent[ x ] x Case 2 Case 3

  23. Insertion into a Red-Black Tree—Summary Observations: ■ Every case takes constant time ■ Case 1 brings x two steps closer to the root and only recolors nodes (no rotations) ■ In Cases 2 and 3, we perform 1 or 2 rotations; then we are done Lemma: An insertion into a red-black tree with n nodes takes O (log n ) time and performs at most two rotations.

  24. Deletion from a Red-Black Tree 17 19 7 13 18 5 37 nil nil nil nil nil 1 49 25 nil nil nil nil nil 33 nil nil Element to delete: 25

  25. Deletion from a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 18 The leaves are black. 5 37 4 nil nil nil nil nil Both children of a red node are black. 1 49 33 nil nil nil nil nil nil Every root-to-leaf path contains the same number of black nodes. Element to delete: 25

  26. Deletion from a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 18 The leaves are black. 5 37 4 nil nil nil nil nil Both children of a red node are black. 1 49 25 nil nil nil nil nil Every root-to-leaf path contains the 33 same number of black nodes. nil nil Element to delete: 18

  27. Deletion from a Red-Black Tree Red-black tree properties 4 Every node has a color, red or black. 17 4 The root is black. 19 7 4 13 nil The leaves are black. 5 37 4 nil nil nil Both children of a red node are black. 1 49 25 nil nil nil nil nil Every root-to-leaf path contains the 8 33 same number of black nodes. nil nil Element to delete: 18

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