SLIDE 1
Red-Black Trees
Binary Search Trees with O(log n) Worst-Case Time per Operation
SLIDE 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. Both children of a red node are black. Every root-to-leaf path contains the same 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.
nil nil nil nil nil nil nil nil nil nil nil nil
SLIDE 3 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. Corollary: A red-black tree with n nodes has height at most 2 log (n + 1).
The Height of a Red-Black Tree with n Nodes
nil nil nil nil nil nil nil nil nil nil nil nil
Lemma: A red-black tree of black-height h contains at least 2h – 1 nodes. Lemma: A red-black tree of black-height h has height at most 2h.
SLIDE 4
Rotations—The Key to Height Reduction
b d A B C a c e b d A B C a c e Right rotation Left rotation
SLIDE 5 A Rotation Takes Constant Time
b d A B C a c e b d A B C a c e
Right rotation
d b c a e
SLIDE 6 A Rotation Takes Constant Time
b d A B C a c e b d A B C a c e
Right rotation
d b c a e
SLIDE 7 A Rotation Takes Constant Time
b d A B C a c e b d A B C a c e
Right rotation
d b c a e
SLIDE 8 A Rotation Takes Constant Time
b d A B C a c e b d A B C a c e
Right rotation
d b c a e
SLIDE 9 A Rotation Takes Constant Time
b d A B C a c e b d A B C a c e
Right rotation
c a e d b
SLIDE 10 nil nil nil nil nil nil nil nil nil nil nil nil
Insertion into a Red-Black Tree
1 5 13 7 17 18 19 37 25 33 49 Element to insert: 21
SLIDE 11 nil nil nil nil nil nil nil nil nil nil nil nil nil
Insertion into a Red-Black Tree
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
4
1 5 13 7 17 18 19 37 25 33 49 Element to insert: 21 21
4 4
SLIDE 12 Insertion into a Red-Black Tree
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
4
Element to insert: 3
4 4
nil nil nil nil nil nil nil nil nil nil nil nil
1 5 13 7 17 18 19 37 25 33 49
SLIDE 13 nil nil nil nil nil nil nil nil nil nil nil nil nil
Insertion into a Red-Black Tree
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
4
Element to insert: 3
4 4
1 5 13 7 17 18 19 37 25 33 49 3
4 8
No red-red property Goal: Restore no-red-red condition while maintaining the other four. Tools: Recoloring of nodes; rotations
SLIDE 14 nil nil nil nil nil nil nil nil nil nil nil nil nil
Insertion into a Red-Black Tree
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
4
Element to insert: 3
4 4
1 5 13 7 17 18 19 37 25 33 49 3
4 8
No red-red property Goal: Restore no-red-red condition while maintaining the other four. Tools: Recoloring of nodes; rotations
SLIDE 15
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;
that is, parent[parent[x]] exists.
■ If parent[x] is black, we are done.
Restoring the No-Red-Red Property
? x parent[x] parent[parent[x]]
SLIDE 16
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;
that is, parent[parent[x]] exists.
■ If parent[x] is black, we are done.
Assumptions:
■ Neither x nor parent[x] is the root ■ parent[x] is the left child of parent[parent[x]]
Restoring the No-Red-Red Property
? x parent[x] parent[parent[x]]
SLIDE 17
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;
that is, parent[parent[x]] exists.
■ If parent[x] is black, we are done.
Assumptions:
■ Neither x nor parent[x] is the root ■ parent[x] is the left child of parent[parent[x]]
Restoring the No-Red-Red Property
x parent[x] parent[parent[x]]
SLIDE 18
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;
that is, parent[parent[x]] exists.
■ If parent[x] is black, we are done.
Assumptions:
■ Neither x nor parent[x] is the root ■ parent[x] is the left child of parent[parent[x]]
Restoring the No-Red-Red Property
x parent[x] parent[parent[x]]
SLIDE 19
x parent[x] parent[parent[x]]
Case 1
Restoring the No-Red-Red Property
3 cases:
■ Both parent[x] and its
sibling are red
■ parent[x] is red, its sibling
is black, and x is the left child of its parent
■ As Case 2; but x is the
right child of its parent x
Recoloring
SLIDE 20
Restoring the No-Red-Red Property
3 cases:
■ Both parent[x] and its
sibling are red
■ parent[x] is red, its sibling
is black, and x is the left child of its parent
■ As Case 2; but x is the
right child of its parent x parent[x] parent[parent[x]] x
Recoloring
x parent[x] parent[parent[x]]
Recoloring Right-rotation Case 1 Case 2
SLIDE 21
Restoring the No-Red-Red Property
3 cases:
■ Both parent[x] and its
sibling are red
■ parent[x] is red, its sibling
is black, and x is the left child of its parent
■ As Case 2; but x is the
right child of its parent x parent[x] parent[parent[x]] x
Recoloring
x parent[x] parent[parent[x]]
Recoloring + Right-rotation Case 2 Case 1
SLIDE 22
Restoring the No-Red-Red Property
3 cases:
■ Both parent[x] and its
sibling are red
■ parent[x] is red, its sibling
is black, and x is the left child of its parent
■ As Case 2; but x is the
right child of its parent x parent[x] parent[parent[x]] x
Recoloring
x parent[x] parent[parent[x]]
Case 2 Recoloring + Right-rotation Left-rotation Case 1 Case 3
SLIDE 23
Lemma: An insertion into a red-black tree with n nodes takes O(log n) time and performs at most two rotations. 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
Insertion into a Red-Black Tree—Summary
SLIDE 24 Deletion from a Red-Black Tree
nil nil nil nil nil nil nil nil nil nil nil nil
1 5 13 7 17 18 19 37 25 33 49 Element to delete: 25
SLIDE 25 Deletion from a Red-Black Tree
1 5 13 7 17 18 19 37 33 49 Element to delete: 25
nil nil nil nil nil nil nil nil nil nil nil
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
4 4 4 4
SLIDE 26 Deletion from a Red-Black Tree
Element to delete: 18 Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
4 4 4 4
nil nil nil nil nil nil nil nil nil nil nil nil
1 5 13 7 17 18 19 37 25 33 49
SLIDE 27 Deletion from a Red-Black Tree
1 5 13 7 17 19 37 25 33 49 Element to delete: 18
nil nil nil nil nil nil nil nil nil nil nil
4 4 4 4
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
8
SLIDE 28 nil nil nil nil nil nil nil nil nil nil nil
Deletion from a Red-Black Tree
1 5 13 7 17 19 37 25 33 49 Element to delete: 18
4 4 4
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes. First step: Declare the child of the removed node “doubly black.”
4 8
SLIDE 29 nil nil nil nil nil nil nil nil nil nil nil
Deletion from a Red-Black Tree
1 5 13 7 17 19 37 25 33 49 Element to delete: 18
4 4 4
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes. First step: Declare the child of the removed node “doubly black.”
4 4
SLIDE 30 nil nil nil nil nil nil nil nil nil nil nil
Deletion from a Red-Black Tree
1 5 13 7 17 19 37 25 33 49 Element to delete: 18
4 4 4 4
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
8
First step: Declare the child of the removed node “doubly black.” Color property
SLIDE 31 nil nil nil nil nil nil nil nil nil nil nil
Deletion from a Red-Black Tree
1 5 13 7 17 19 37 25 33 49 Element to delete: 18
4 4 4 4
Red-black tree properties Every node has a color, red or black. The root is black. The leaves are black. Both children of a red node are black. Every root-to-leaf path contains the same number of black nodes.
8
First step: Declare the child of the removed node “doubly black.” Color property
SLIDE 32
Restoring the Color Property
The easy cases: Node x is red, root or not Node x is the root and black
SLIDE 33 x parent[x] w c c'
Case 2
x parent[x] w c
Case 1
Restoring the Color Property
Assume that x is the left child of its parent and that its right sibling w is black. 3 cases, depending on colors
■ Both are black ■ Right child is red ■ Left child is red, and right
child is black
Recoloring
x c c c'
Recoloring
c c'
Left-rotation
SLIDE 34 Case 3
x parent[x] w c
Recoloring
Restoring the Color Property
Assume that x is the left child of its parent and that its right sibling w is black. 3 cases, depending on colors
■ Both are black ■ Right child is red ■ Left child is red, and right
child is black x parent[x] w c x c
Case 1 Recoloring
c c
Right-rotation Case 2
SLIDE 35
Case 4
x w parent[x]
Restoring the Color Property
The missing case: The right sibling w of x is red. Observations:
■ Cases 2 and 3 perform at most two rotations; then we’re done ■ Case 1 only recolors and brings us one step closer to the root ■ Case 4 performs one rotation and bring us one step away from the root! ■ But then Cases 2 and 3 finish as before ■ If Case 1 applies after the rotation, the next iteration exits because parent[x] is
red Lemma: A deletion from a red-black tree with n nodes takes O(log n) time and performs at most three rotations. x w (new) w (old)
Left-rotation
x w parent[x]
Recoloring
SLIDE 36
Red-black trees support all dictionary operations in O(log n) time:
■ Insert ■ Delete ■ Find ■ Predecessor ■ Successor ■ Minimum ■ Maximum
To re-balance the tree after an update, only O(1) rotations are necessary. The latter will be handy for data structures that store more complicated information at every node, which is more expensive to restore after a rotation.
Red-Black Trees—Summary
