Red-Black Trees 1© 2004 Goodrich, Tamassia
Red-Black Trees
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3 8
4
v
z
Red-Black Trees 2© 2004 Goodrich, Tamassia
Recall: Representing general trees as binary trees
A
B D
F
Supplementary:
C
EC
A
B
DF
CE
C
Left child pointer points to a child node, right child pointer points to a sibling node
Red-Black Trees 3© 2004 Goodrich, Tamassia
From (2,4) to Red-Black TreesA red-black tree is a representation of a (2,4) tree by means of a binary tree whose nodes are colored red or blackIn comparison with its associated (2,4) tree, a red-black tree hasn same logarithmic time performancen simpler implementation with a single node type
2 6 73 54
4 6
2 7
5
3
3
5OR
Red-Black Trees 4© 2004 Goodrich, Tamassia
Red-Black Trees (§ 9.5)A red-black tree can also be defined as a binary search tree that satisfies the following properties:n Root Property: the root is blackn External Property: every leaf is blackn Internal Property: the children of a red node are blackn Depth Property: all the leaves have the same black depth
9
154
62 12
7
21
Red-Black Trees 5© 2004 Goodrich, Tamassia
Height of a Red-Black TreeTheorem: A red-black tree storing n entries has height O(log n)Proof:n The height of a red-black tree is at most twice the height of
its associated (2,4) tree, which is O(log n)
The search algorithm for a binary search tree is the same as that for a binary search treeBy the above theorem, searching in a red-black tree takes O(log n) time
Red-Black Trees 6© 2004 Goodrich, Tamassia
InsertionTo perform operation insert(k, o), we execute the insertion algorithm for binary search trees and color red the newly inserted node z unless it is the rootn We preserve the root, external, and depth propertiesn If the parent v of z is black, we also preserve the internal property and
we are done n Else (v is red ) we have a double red (i.e., a violation of the internal
property), which requires a reorganization of the treeExample where the insertion of 4 causes a double red:
6
3 8
6
3 8
4z
v v
z
Red-Black Trees 7© 2004 Goodrich, Tamassia
Remedying a Double RedConsider a double red with child z and parent v, and let w be the sibling of v
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6
7z
vw2
4 6 7
.. 2 ..
Case 1: w is blackn The double red is an incorrect
replacement of a 4-noden Restructuring: we change the
4-node replacement
Case 2: w is redn The double red corresponds
to an overflown Recoloring: we perform the
equivalent of a split
4
6
7z
v
2 4 6 7
2w
Red-Black Trees 8© 2004 Goodrich, Tamassia
RestructuringA restructuring remedies a child-parent double red when the parent red node has a black siblingIt is equivalent to restoring the correct replacement of a 4-nodeThe internal property is restored and the other properties are preserved
4
6
7z
vw2
4 6 7
.. 2 ..
4
6
7
z
v
w2
4 6 7
.. 2 ..
ba cba
c
aa bb cc
Supplementary:a, b, c
Red-Black Trees 9© 2004 Goodrich, Tamassia
Restructuring (cont.)There are four restructuring configurations depending on whether the double red nodes are left or right children
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4
6
6
2
4
6
4
2
2
6
4
2 6
4
Supplementary:a, b, c, d
ba c d
b
a
c
db
a
c
d
ba
cd
ba
c d
Red-Black Trees 10© 2004 Goodrich, Tamassia
RecoloringA recoloring remedies a child-parent double red when the parent red node has a red siblingThe parent v and its sibling w become black and the grandparent ubecomes red, unless it is the rootIt is equivalent to performing a split on a 5-nodeThe double red violation may propagate to the grandparent u
4
6
7z
v
2 4 6 7
2w 4
6
7z
v
6 7
2w
… 4 …
2
Red-Black Trees 11© 2004 Goodrich, Tamassia
Analysis of InsertionRecall that a red-black tree has O(log n) heightStep 1 takes O(log n) time because we visit O(log n)nodesStep 2 takes O(1) timeStep 3 takes O(log n) time because we performn O(log n) recolorings, each
taking O(1) time, andn at most one restructuring
taking O(1) time
Thus, an insertion in a red-black tree takes O(log n) time
Algorithm insert(k, o)
1. We search for key k to locate the insertion node z
2. We add the new entry (k, o) at node z and color z red
3. while doubleRed(z)if isBlack(sibling(parent(z)))
z ← restructure(z)return
else { sibling(parent(z) is red }z ← recolor(z)
Red-Black Trees 12© 2004 Goodrich, Tamassia
DeletionTo perform operation remove(k), we first execute the deletion algorithm for binary search treesLet v be the internal node removed, w the external node removed, and r the sibling of wn If either v of r was red, we color r black and we are donen Else (v and r were both black) we color r double black, which is a
violation of the internal property requiring a reorganization of the tree
Example where the deletion of 8 causes a double black:
6
3 8
4
v
r w
6
3
4
r
Red-Black Trees 13© 2004 Goodrich, Tamassia
Remedying a Double BlackThe algorithm for remedying a double black node w with sibling y considers three casesCase 1: y is black and has a red childn We perform a restructuring, equivalent to a transfer , and we are
doneCase 2: y is black and its children are both blackn We perform a recoloring, equivalent to a fusion, which may
propagate up the double black violationCase 3: y is redn We perform an adjustment, equivalent to choosing a different
representation of a 3-node, after which either Case 1 or Case 2 applies
Deletion in a red-black tree takes O(log n) time
Red-Black Trees 14© 2004 Goodrich, Tamassia
Red-Black Tree Reorganizationremedy double redInsertion
double red removed or propagated up
splitrecoloring
double red removedchange of 4-node representationrestructuring
result(2,4) tree actionRed-black tree action
remedy double blackDeletion
restructuring or recoloring follows
change of 3-node representation
adjustment
double black removed or propagated upfusionrecoloring
double black removedtransferrestructuring
result(2,4) tree actionRed-black tree action
Red-Black Trees 15© 2004 Goodrich, Tamassia
Why Red-Black trees if we already have AVL-trees and 2-4 trees?
Red-Black trees, AVL trees and 2-4 trees all assure logarithmic height and search time
AVL trees may require a logarithmic number of rotations or “pointer dances” per insertion or deletion.
2-4 trees my require a logarithmic number of split or merge operations per insertion or deletion
Red-Black trees get away with only a constant number of restructures, or adjustments (“pointer dances”) per insertion ordeletion and a logarithmic number of recolorings which are much simpler
java.util uses red-black trees for its tree-based containers
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