Red Black Trees

1
Red Black Trees

• Deletion
• (Last time Insertion get notes if you dont
  have them)

2
Deletion from Red Black Tree

• Works exactly like deletion from Binary Search
  Tree (BST)
• UP TO fixing Red-Black tree
• Steps
• Find element (x) to delete
• Swap element with leftmost right
• Delete node that now contains element (x)
• Reattach right child

3
BST delete
51
54
19
88
19
88
71
71
83
54
83
60
55
60
51
51
55
4
R-B Tree Delete

• Find
• Swap
• Moves entry to node with one external node (left)
• Remove entry
• Reattach right child

5
R-B Tree Delete

• What about color?
• Deleting node can introduce
• red parent, red child problem
• Black depth problem

6
R-B Delete Case 1

• Red node deleted
• Entry swapped into red node, red node deleted
• Problems?
• Can this lead to red parent, red child problem?
• Can this lead to black depth problem?

7
R-B Delete Case 2

• Delete black node with red right child
• Left child was dummy placeholder (black)
• Problems?
• Red parent, red child?
• Black depth?

8
R-B Delete Case 2

• Could have either problem
• Solution?
• Recolor child of deleted node to black

9
R-B Delete Case 3

• Delete black node with black child
• Problem
• Cant recolor to fix black depth
• Black depth is broken

10
R-B Delete Case 3

• Black depth is broken need to fix
• Cant fix with recoloring
• Color R with fictitious color
• double black
• Simply way of tracking where violated
• black depth
• How we fix double black problem
• depends on neighboring nodes

X
V
R
X
R
11
Double Black in 2-4 Tree

• Like having underflow in 2-4 Tree

X
X
2-4
R
R
12
Recap of Delete

• 3 Cases
• Red node deleted
• No problem
• Delete black node with red child
• Color red child black
• Delete black node with black child
• Color child Double Black
• 3 possibilities depending on neighboring nodes
• Rs sibling is black with at least one red child
• Rs sibling is black with no red children
• Rs sibling is red

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R-B Delete Case 3a

• Rs sibling is black with at least one red child
• Goal reintroduce black node in path that
  contains R
• Equivalent to transferring (rotation) in 2-4 Tree
• Move parent down, sibling or red child of former
  sibling moves into parents node

14
R-B Delete Case 3a Example
15
R-B Delete Case 3b
X
Think of how this looks as 2-4 Tree What do we
do to the 2-4 Tree?
Y
R
S
Z
In 2-4 we drop parent and propagate underflow
upward
X
Y
R
S
Z
16
R-B Delete Case 3b
X
Think of how this looks as 2-4 Tree What do we
do to the 2-4 Tree?
Y
R
S
Z
In 2-4 we drop parent and propagate underflow
upward BUT X was in red node it was part of a 3
node so underflow goes no further
X
Y
R
S
Z
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R-B Delete Case 3b Summary

• When sibling of double black node is black
  without red children
• Pass double black up tree, like passing underflow
  up in 2-4 tree
• If double black reaches root, color root single
  black, decreasing black depth by one for entire
  tree

18
R-B Delete Case 3c

• Rs sibling is red
• Fix by rotating
• Rs sibling will be black (in new location!)
• Rs parent red (in new location!)
• R still double black
• Use one of previous cases (3a or 3b)

19
R-B Delete Case 3c Example
20
R-B Delete Example
21
Run-Times

• Search Trees Find, Insert, Next, Previous
• BST O(h)
• 2-4 Tree O(log n)
• R-B Tree O(log n)
• All operations are O(h) h varies