Analysis of Algorithms CS 477/677

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Analysis of AlgorithmsCS 477/677

• Red-Black Trees
• Instructor George Bebis
• (Chapter 14)

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Red-Black Trees

• Balanced binary search trees guarantee an
  O(lgn) running time
• Red-black-tree
• Binary search tree with an additional attribute
  for its nodes color which can be red or black
• Constrains the way nodes can be colored on any
  path from the root to a leaf
• Ensures that no path is more than twice as long
  as any other path ? the tree is balanced

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Example RED-BLACK-TREE
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17
41
NIL
NIL
30
47
38
50
NIL
NIL
NIL
NIL
NIL
NIL

• For convenience we use a sentinel NILT to
  represent all the NIL nodes at the leafs
• NILT has the same fields as an ordinary node
• ColorNILT BLACK
• The other fields may be set to arbitrary values

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Red-Black-Trees Properties

• (Satisfy the binary search tree property)
• Every node is either red or black
• The root is black
• Every leaf (NIL) is black
• If a node is red, then both its children are
  black
• No two consecutive red nodes on a simple path
• from the root to a leaf
• For each node, all paths from that node to
  descendant leaves contain the same number of
  black nodes

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Black-Height of a Node

• Height of a node the number of edges in the
  longest path to a leaf
• Black-height of a node x bh(x) is the number of
  black nodes (including NIL) on the path from x to
  a leaf,
• not counting x

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Most important property of Red-Black-Trees

• A red-black tree with n internal nodes
• has height at most 2lg(n 1)
• Need to prove two claims first

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Claim 1

• Any node x with height h(x) has bh(x) h(x)/2
• Proof
• By property 4, at most h/2 red nodes on the path
  from the node to a leaf
• Hence at least h/2 are black

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Claim 2

• The subtree rooted at any node x contains at
  least 2bh(x) - 1 internal nodes

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Claim 2 (contd)

• Proof By induction on hx
• Basis hx 0 ?
• x is a leaf (NILT) ?
• bh(x) 0 ?
• of internal nodes 20 - 1 0
• Inductive Hypothesis assume it is true for
  hxh-1

x
NIL
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Claim 2 (contd)

• Inductive step
• Prove it for hxh
• Let bh(x) b, then any child y of x has
• bh (y)
• bh (y)

b (if the child is red), or
b - 1 (if the child is black)
NIL
NIL
NIL
NIL
NIL
NIL
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Claim 2 (contd)

• Using inductive hypothesis, the number of
  internal nodes for each child of x is at least
• 2bh(x) - 1 - 1
• The subtree rooted at x contains at least
• (2bh(x) - 1 1) (2bh(x) - 1 1) 1
• 2 (2bh(x) - 1 - 1) 1
• 2bh(x) - 1 internal nodes

h
h-1
bh(l)bh(x)-1 bh(r)bh(x)-1
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Height of Red-Black-Trees (contd)

• Lemma A red-black tree with n internal nodes has
  height at most 2lg(n 1).
• Proof
• n
• Add 1 to both sides and then take logs
• n 1 2b 2h/2
• lg(n 1) h/2 ?
• h 2 lg(n 1)

root
height(root) h
bh(root) b
r
l
2b - 1
2h/2 - 1
number n of internal nodes
since b ? h/2
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Operations on Red-Black-Trees

• The non-modifying binary-search-tree operations
  MINIMUM, MAXIMUM, SUCCESSOR, PREDECESSOR, and
  SEARCH run in O(h) time
• They take O(lgn) time on red-black trees
• What about TREE-INSERT and TREE-DELETE?
• They will still run on O(lgn)
• We have to guarantee that the modified tree will
  still be a red-black tree

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INSERT

• INSERT what color to make the new node?
• Red? Lets insert 35!
• Property 4 is violated if a node is red, then
  both its children are black
• Black? Lets insert 14!
• Property 5 is violated all paths from a node to
  its leaves contain the same number of black nodes

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DELETE

• DELETE what color was the
• node that was removed? Black?
• Every node is either red or black
• The root is black
• Every leaf (NIL) is black
• If a node is red, then both its children are
  black
• For each node, all paths from the node to
  descendant leaves contain the same number of
  black nodes

OK!
OK!
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Rotations

• Operations for re-structuring the tree after
  insert and delete operations on red-black trees
• Rotations take a red-black-tree and a node within
  the tree and
• Together with some node re-coloring they help
  restore the red-black-tree property
• Change some of the pointer structure
• Do not change the binary-search tree property
• Two types of rotations
• Left right rotations

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Left Rotations

• Assumptions for a left rotation on a node x
• The right child of x (y) is not NIL
• Idea
• Pivots around the link from x to y
• Makes y the new root of the subtree
• x becomes ys left child
• ys left child becomes xs right child

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Example LEFT-ROTATE
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LEFT-ROTATE(T, x)

1. y ? rightx ?Set y
2. rightx ? lefty ? ys left subtree becomes
   xs right subtree
3. if lefty ? NIL
4. then plefty ? x ? Set the parent relation
   from lefty to x
5. py ? px ? The parent of x becomes the
   parent of y
6. if px NIL
7. then rootT ? y
8. else if x leftpx
9. then leftpx ? y
10. else rightpx ? y
11. lefty ? x ? Put x on ys left
12. px ? y ? y becomes xs parent

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Right Rotations

• Assumptions for a right rotation on a node x
• The left child of y (x) is not NIL
• Idea
• Pivots around the link from y to x
• Makes x the new root of the subtree
• y becomes xs right child
• xs right child becomes ys left child

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Insertion

• Goal
• Insert a new node z into a red-black-tree
• Idea
• Insert node z into the tree as for an ordinary
  binary search tree
• Color the node red
• Restore the red-black-tree properties
• Use an auxiliary procedure RB-INSERT-FIXUP

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RB Properties Affected by Insert
OK!

• Every node is either red or black
• The root is black
• Every leaf (NIL) is black
• If a node is red, then both its children are
  black
• For each node, all paths
• from the node to descendant
• leaves contain the same number
• of black nodes

If z is the root ? not OK
OK!
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RB-INSERT-FIXUP Case 1

• zs uncle (y) is red
• Idea (z is a right child)
• ppz (zs grandparent) must be black z and
  pz are both red
• Color pz black
• Color y black
• Color ppz red
• z ppz
• Push the red violation up the tree

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RB-INSERT-FIXUP Case 1

• zs uncle (y) is red
• Idea (z is a left child)
• ppz (zs grandparent) must be black z and
  pz are both red
• color pz ? black
• color y ? black
• color ppz ? red
• z ppz
• Push the red violation up the tree

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RB-INSERT-FIXUP Case 3

• Case 3
• zs uncle (y) is black
• z is a left child

• Idea
• color pz ? black
• color ppz ? red
• RIGHT-ROTATE(T, ppz)
• No longer have 2 reds in a row
• pz is now black

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RB-INSERT-FIXUP Case 2

• Case 2
• zs uncle (y) is black
• z is a right child
• Idea
• z ? pz
• LEFT-ROTATE(T, z)
• ? now z is a left child, and both z and pz are
  red ? case 3

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RB-INSERT-FIXUP(T, z)

1. while colorpz RED
2. do if pz leftppz
3. then y ? rightppz
4. if colory RED
5. then Case1
6. else if z rightpz
7. then Case2
8. Case3
9. else (same as then clause with
   right and left
   exchanged)
10. colorrootT ? BLACK

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Example
Insert 4
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y
z
y
z and pz are both red zs uncle y is red
z and pz are both red zs uncle y is black z is
a right child
z
z and pz are red zs uncle y is black z is a
left child
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RB-INSERT(T, z)

1. y ? NIL
2. x ? rootT
3. while x ? NIL
4. do y ? x
5. if keyz lt keyx
6. then x ? leftx
7. else x ? rightx
8. pz ? y

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RB-INSERT(T, z)

1. if y NIL
2. then rootT ? z
3. else if keyz lt keyy
4. then lefty ? z
5. else righty ? z
6. leftz ? NIL
7. rightz ? NIL
8. colorz ? RED
9. RB-INSERT-FIXUP(T, z)

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Analysis of RB-INSERT

• Inserting the new element into the tree O(lgn)
• RB-INSERT-FIXUP
• The while loop repeats only if CASE 1 is executed
• The number of times the while loop can be
  executed is O(lgn)
• Total running time of RB-INSERT O(lgn)

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Red-Black Trees - Summary

• Operations on red-black-trees
• SEARCH O(h)
• PREDECESSOR O(h)
• SUCCESOR O(h)
• MINIMUM O(h)
• MAXIMUM O(h)
• INSERT O(h)
• DELETE O(h)
• Red-black-trees guarantee that the height of the
  tree will be O(lgn)

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Problems

• What is the ratio between the longest path and
  the shortest path in a red-black tree?
• - The shortest path is at least bh(root)
• - The longest path is equal to h(root)
• - We know that h(root)2bh(root)
• - Therefore, the ratio is 2

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Problems

• What red-black tree property is violated in the
  tree below? How would you restore the red-black
  tree property in this case?
• Property violated if a node is red, both its
  children are black
• Fixup color 7 black, 11 red, then right-rotate
  around 11

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Problems

• Let a, b, c be arbitrary nodes in subtrees ?, ?,
  ? in the tree below. How do the depths of a, b, c
  change when a left rotation is performed on node
  x?
• a increases by 1
• b stays the same
• c decreases by 1

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Problems

• When we insert a node into a red-black tree, we
  initially set the color of the new node to red.
  Why didnt we choose to set the color to black?
• (Exercise 13.4-7, page 294) Would inserting a new
  node to a red-black tree and then immediately
  deleting it, change the tree?