Chapter 13 Red-Black Trees Lee, Hsiu-Hui Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the

Chapter 13

Red-Black Trees Lee, Hsiu-Hui

Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the web.

20071130 chap13 Hsiu-Hui Lee 2

• Many search-tree schemes that are “balanced” in order to guarantee that basic dynamic-set operations take O(lg n ) time in the worse case.

e.g. AVL trees Red-black trees

Balanced search Trees


Red-Black Tree

• a variation of binary search trees• with one extra bit of storage per node: its color, which

can either RED or BLACK. • Each node of the tree now contains the fields color, key,

left, right, and p. If a child or the parent of a node does not exist, the corresponding pointer field of the node contains the value NIL. We shall regard these NIL’s as being pointers to external nodes(leaves) of the binary search tree and the normal, key-bearing nodes as being internal nodes of the tree.


Red-Black Properties

1. Every node is either red or black.2.2. The root is black.The root is black.3. Every leaf (NIL) is black4. If a node is red, then both its children are black.5. For each node, all paths from the node x to desce

ndant leaves contain the same number of black nodes (i.e. black-height(x)).


Example of a red-black tree



1. Every node is either red or black.



2. 3. The root and leaves (NIL’s) are black.


4. If a node is red, then its children are black.

?? Means (If a node is red, then its parent is black.)



5. All simple paths from any node x to a descendant leaf have the same number of black nodes = black-height(x).



black-height of the node: bh(x)

# of black nodes on any path from, but not including,

a node x down to a leaf

black-height of a RB tree = black-height of its root



Leaves and the root’s parent omitted entirely


Height and black-height

• Height of a node x: h(x)

is the number of edges in a longest path to a leaf.• Black-height of a node x: bh(x) is the number of black nodes (including nil[T ]) on the path from x to leaf, not counting x.


Lemma 13.0aThe subtree rooted at any node x contains ≥ 2bh(x) − 1 internal nodes.Proof By induction on height of x.Basis: Height of x = 0 ⇒ x is a leaf bh⇒ (x) = 0. The subtree

rooted at x has 0 internal nodes. 20 − 1 = 0.Inductive step: Let the height of x be h and bh(x) = b. Any child of x has height h − 1 and black-height either b

(if the child is red) or b − 1 (if the child is black). By the inductive hypothesis, each child has ≥ 2bh(x)-1 − 1 in

ternal nodes.Thus, the subtree rooted at x contains ≥ 2 · (2bh(x)-1 − 1 )+1 =

2bh(x) −1 internal nodes. (The +1 is for x itself.)


Lemma 13.0bAny node with height h has black –height at

least h/2.

Proof By property 4 (If a node is red, then both its

children are black), ≤ h/2 nodes on the path from the node to a leaf are

red.Hence ≥ h/2 are black.


Lemma 13.1A red-black tree with n internal nodes h

as height at most 2lg(n+1).Proof

Let h and b be the height and black-height of the root, respectively.

By the above two claims,

n ≥ 2b − 1 ≥ 2h/2 − 1 .

Adding 1 to both sides and then taking logs gives

lg(n + 1) ≥ h/2,

which implies that h ≤ 2 lg(n + 1).


13.2 Rotations

Left-Rotation makes y’s left subtree into x’s right subtree.Right-Rotation makes x’s right subtree into y’s left subtree.


LEFT-ROTATE(T, x)

1. y ← right[x] ✄ Set y.

2. right[x] ← left[y] ✄ Turn y’s left subtree into x’s right subtree.

3. if left[y] ≠ nil[T ]4. then p[left[y]] ← x5. p[y] ← p[x] ✄ Link x’s parent to y.

6. if p[x] = nil[T ]7. then root[T ]← y8. else if x = left[p[x]]9. then left[p[x]] ← y10. else right[p[x]] ← y11. left[y] ← x ✄ Put x on y’s left.

12. p[x] ← y

The code for RIGHT-ROTATE is symmetric.Both LEFT-ROTATE and RIGHT-ROTATErun in O(1) time.


Example of LEFT-ROTATE(T,x)Rotation makes y’s left subtree into x’s right subtree.


Some Good Java Appletsto simulate Red-Black Tree• http://

webpages.ull.es/users/jriera/Docencia/AVL/AVL tree applet.htm

• http://gauss.ececs.uc.edu/RedBlack/redblack.html

Insertion and Deletion


RB-INSERT(T, x)1. y ← nil[T]2. x ← root[T]3. while x ≠ nil[T]4. do y ← x5. if key[z] < key[x]6. then x ← left[x]7. else x ← right[x]8. p[z] ← y9. if y = nil[T]10. then root[T] ← z11. else if key[z] < key[y]12. then left[y] ← z13. else right[y] ← z14. left[z] ← nil[T]15. right[z] ← nil[T]16. color[z] ← RED17. RB-INSERT-FIXUP(T, z)


RB-INSERT-FIXUP(T, z)1 while color[p[z]] = RED

2 do if p[z] = left[p[p[z]]]

3 then y ← right[p[p[z]]]

4 if color[y] = RED

5 then color[p[z]] ←BLACK Case 1

6 color[y] ← BLACK Case 1

7 color[p[p[z]]] ← RED Case 1

8 z ← p[p[z]] Case 1


9 else if z = right[p[z]]

10 then z ← p[p[z]] Case 2

11 LEFT-ROTATE(T,z) Case 2

12 color[p[z]] ← BLACK Case 3

13 color[p[p[z]]] ← RED Case 3

14 RIGHT-ROTATE(T,p[p[z]]) Case 3

15 else (same as then clause

with “right” and “left” exchanged)

16 color[root[T]] ← BLACK


The operation of RB-INSERT-FIXUP

紅叔

黑叔右子


黑叔左子


case 1 of the RB-INSERT(z’s uncle y is red)

紅叔右子

紅叔左子


case 2/3 of the RB-INSERT(z’s uncle y is black and z is a right/left child)

黑叔右子黑叔左子


Analysis

• RB-INSERT take a total of O(lg n) time.• It never performs more than two rotatio

ns, since the while loop terminates if case 2 or case 3 executed.


13.4 Deletion

RB-DELETE(T, z)1 if left[z] = nil[T] or right[z] = nil[T]

2 then y ← z

3 else y ← TREE-SUCCESSOR(z)

4 if left[y] ≠ nil[T]

5 then x ← left[y]

6 else x ← right[y]

7 p[x] ← p[y]

8 if p[y] = nil[T]


9 then root[T] ← x10 else if y = left[p[y]]11 then left[p[y]] ← x12 else right[p[y]] ← x13 if y ≠ z14 then key[z] ← key[y]15 copy y’s satellite data into z16 if color[y] = BLACK

17 then RB-DELETE-FIXUP(T, x)18 return y


RB-DELETE-FIXUP(T,x)

RB-DELETE-FIXUP (T, x)

1 while x ≠ root[T] and color[x] = BLACK

2 do if x = left[p[x]]

3 then w ← right[p[x]]

4 if color[w] = RED

5 then color[w] ← BLACK Case1

6 color[p[x]] = RED Case1

7 LEFT-ROTATE(T,p[x]) Case1

8 w ← right[p[x]] Case1


9 if color[right[w]] = BLACK and color[right[w]= BLACK10 then color[w] ← RED Case211 x ← p[x] Case212 else if color[right[w]] = BLACK 13 then color[left[w]] ← BLACK Case314 color[w] ← RED Case315 RIGHT-ROTATE(T,w) Case3 16 w ← right[p[x]] Case317 color[w] ← color[p[x]] Case418 color[p[x]] ← BLACK Case419 color[right[w]] ← BLACK Case420 LEFT-ROTATE(T,p[x]) Case421 x ← root[T] Case422 else (same as then clause with “right” and “left” exchan

ged)23 color[x] ← BLACK


the case in the while loop of RB-DELETE-FIXUP



Analysis

• The RB-DELETE-FIXUP takes O(lg n) time and performs at most three rotations.

• The overall time for RB-DELETE is therefore also O(lg n)

Documents

Chapter 13 Red-Black Trees Lee, Hsiu-Hui Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the