CS473-Algorithms

1
CS473-Algorithms

• Lecture
• RED-BLACK TREES (RBT)

2
Overview

• Previous lecture showed that a BST of height h
  can implement any of the dynamic set operations
  in O(h) time.
• Thus, set operations are fast if the height of
  BST is small.
• But, if h is large, its performance is no better
  than linked list.

3
Overview

• Red-Black trees are one of many search tree
  schemes that are balanced to guarantee h O(lgn)
• First, we will show that h O(lgn) in RBTs
• Then, show how to manipulate maintain RBTs
  during dynamic set operations.

4
Properties of Red-Black Trees

• Add color field (1 bit) to each node of a BST.
• Consider NIL child fields as pointer to
  external nodes (leafs)
• i.e. Leaf nodes do not have keys
• Consider normal key-bearing nodes as internal
  nodes.
• Hence, each internal node has exactly 2 children.
• Thus, RBTs are full binart trees (FBTs)
• FBT Each node is either a leaf or has degree
  exactly 2.
• i.e. There are no degree-1 nodes.

5
Red-Black Properties

• Every node is either red or black
• Every leaf (NIL) is black
• If a node is red, then both of its children are
  black.
• i.e. There cant be 2 consecutive reds on a
  simple path.
• Every simpe path from a node to a descendant leaf
  contains the same number of black nodes.
• ( Additional propery) The root is black.
• Simplifies explanation of insertion which assumes
  and guarantees that the root is black.

6
A Sample Red Black Tree
bh 2
h 4
bh 1
bh 2
h 1
h 3
bh 1
bh 2
h 2
h 4
bh 1
h 1
bh 0
h 0
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Properties of Red-Black Trees

• Black-Height bh (x)
• of blacks on the simple paths to all descendant
  leaves not counting x.
• By the property 4, the black-height is well
  defined.
• Red-Black properties ensure that
• No simpe path from a node to a descendant leaf
  is more that twice as long as any other.
• So, red-black trees are apptoximately balanced.

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Properties of Red-Black Trees

• Lemma 1 A height h node has black height h/2
• Height is length of the longest path to a leaf.
• By property 3, of reds on this path h/2
• Hence, of blacks h/2
• Lemma 2 The subtree Tx rooted at any node x
  contains at least Tx 2bh(x) 1 internal
  nodes.

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Properties of Red-Black Trees

• Basis h(x)0 x must be a leaf (NIL)
  bh(x)0
• Tx contains 0 internal nodes 2bh(x) -12-1 0
• Inductive Step h(x) gt 0 x is internal node
  with 2 children
• lx leftx
• rx rightx

x
h(x)
bh(x)
rx
lx
Trx
Tlx
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Properties of Red-Black Trees

• Depending on lx and rx being red or black
• bh(lx) bh(x) or bh(x) -1, respectively,
• bh(rx) bh(x) or bh(x) -1, respectively.
• Since h(lx), h(rx) lt h(x)
• we can apply inductive hypothesis.
• Tx Tl Tr 1
• (2bh(x)-1 1) (2bh(x)-1 1) 1
• 2bh(x)-1 1 Q.E.D.

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Properties of Red-Black Trees

• Theorem A red-black tree with n internal nodes
  has height at most 2lg(n1)
• Due to lemma 2
• Troot n 2bh(root) 1
• Due to lemma 1
• bh(root) h(root) / 2 H / 2
• n 2H/2 -1
• (n1) 2H/2
• H 2 lg(n1)

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Properties of Red-Black Trees

• Corollary Dynamic set operations MIN, MAX,
  SEARCH, SUCCESSOR, PREDECESSOR take O(lgn) time.
• Fact Let lmin(x) lmax(x) denote the lengths of
  the shortest longest paths from a node x to its
  leafs, respectively.
• Then, lmax(x) lmin(x) for any node x.
• Proof lmax(x) h(x) 2bh(x) lmin(x)

Lemma 1
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The Problem of Insertion into Red-Black Trees

• Draw a R-B tree
• Color choices starting at 12
• Forced choices 12 ? R 5 and 9 ? B

7
5
9
12
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The Problem of Insertion into Red-Black Trees

• Insert 8 (Left child of 9)
• No problem, Just color it red

7
5
9
8
? ? R
12
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The Problem of Insertion into Red-Black Trees

• Insert 11 (Left child of 12)
• 11 cant be red (Violate property 3)
• 11 cant be black (Violate property 4)

7
5
9
8
12
?
11
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The Problem of Insertion into Red-Black Trees

• Can fix up by recoloring the tree
• New 11 to Red
• Change 9 to Red
• Change 8 and 12 to Black

7
5
9
B ? R
8
R ? B
R ? B
12
??R
11
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The Problem of Insertion into Red-Black Trees

• Insert 10 (Left child of 11)
• Recoloring is not enough, why?
• -Because of tree imbalance
• -Cant satisfy property 4 (Equal of blacks on
  paths)
• without violating property 3 (No consecutive
  reds on paths)
• Must change tree structure
• -Goal Restructure in O(lgn) time.

7
5
9
8
12
11
?
10