CSE - 5311 Advanced Algorithms

1
CSE - 5311 Advanced Algorithms

• Instructor Gautam Das
• Submitted by
• Raja Rajeshwari Anugula Srujana Tiruveedhi

2
Contents

• RED BLACK Trees
• History
• Properties
• Rotations
• Insertion
• Hashing
• Union Find Algorithm

3
RED BLACK TREES

• RB trees is a Binary Tree with one extra bit of
  storage per node its color, which can be either
  RED or BLACK.
• Each node of the tree contains fields color, key,
  left, right, parent.
• Red-Black Trees ensure that longest path is no
  more than twice as long as any other path so that
  the tree is approximately BALANCED.

4
RED BLACK TREES

• STRUCTURAL PROPERTIES
• Every node is colored red or black.
• The Root is Black.
• Every leaf (Nil) is colored black.
• Both children of a red node are black.
• Every simple path from a child of node X to a
  leaf has the same number of black nodes.

5
RED BLACK TREES

• Points to remember
• This number is known as the black-height of
  X(bh(X)).
• A RB tree with n internal nodes has a height of
  almost 2log(n1).
• Maximum path length is O(log n).
• Finding an element is real quick in RB trees,
  i.e,., it takes O(log n) time.
• Insertion and Deletion take O(log n) time.

6
RED BLACK TREES

• ROTATIONS
• Insertion and Deletion modify the tree, the
  result may violate the properties of red black
  trees. To restore these properties rotations are
  done.
• We can have either LEFT rotation or RIGHT
  rotation by which we must change colors of some
  of the nodes in the tree and also change the
  pointer structure.

7
RED BLACK TREES

• LEFT ROTATE
• RIGHT ROTATE
  
• 
  
  
• a
  
  c
• 
  
• 
  
• a
  b
• b c
• When we do a LEFT rotation on a node x we assume
  that its right child y is not nil ,i.e x may be
  any node in the tree whose right child is not
  Nil.
• It makes y the new root of the sub tree, with x
  as ys left child and ys left child as xs right
  child.

8
RED BLACK TREES

• The Idea to insert is that we traverse the tree
  to see where it fits , assuming that it fits at
  the end, and initially color the inserted node
  RED, then traverse up again.
• Coloring rule while insertion
• If the father node and uncle node of the inserted
  node are red then make father and uncle as BLACK
  and grand father as RED.

9
RED BLACK TREES

• Case 1a Father and Uncle are Red , problem node
  is right child

C
C
Recolor it to BLACK
After Recoloring
D
A
D
A
a
d
e
e
B
B
d
a
b
c
b
c
10
RED BLACK TREES

• Case 1b Father and Uncle are Red , problem node
  is left child

C
Recolor it to BLACK
C
After Recoloring
B
D
B
D
c
A
c
d
e
d
e
A
a
b
a
b
11
RED BLACK TREES
Case 2a Father red and Uncle Black, problem node
is left child
B
C
After Rotation
D
C
B
A
d
c
A
c
b
a
D
a
b
d
12
RED BLACK TREES
Case 2b Father red and Uncle Black, problem node
is right child



C
C
After Rotation
D
A
B
D
d
B
d
c
A
a
c
b
b
a
apply 2a for the above tree


13
RED BLACK TREES

Example

11
11
Apply Case 1b
2
14
2
14
7
15
7
15
1
1
8
5
8
5
Insert Node 4
4
4
14
RED BLACK TREES


11
11
Apply Case 2b
2
14
7
14
7
15
8
15
1
2
8
5
5
1
4
4
15
RED BLACK TREES


11
7
Apply Case 2a
7
14
2
11
8
15
14
5
2
8
1
15
4
5
1
4
16
HASHING

• Has table is an effective data structure for
  implementing dictionaries.
• Although searching for an element in hash table
  in the worst case is T(n) time, under reasonable
  assumptions the expected time to search for an
  element is O(1).
• With hashing this element is stored in slot h(k)
• i.e we use a hash function h to compute the
  slot from the key k. (h maps the universe U of
  keys into the slots of a hash table T0m-1)
• h U 0,1,2..m-1
• Two keys may hash to the same slot. This is
  called collision.

17
UNION FIND

• Basics
• Applications involve grouping n elements into a
  collection of disjoint sets.
• The important operations are
• MAKE-SET(x) Creates a new set
• UNION(x,y) Unites the dynamic sets that contain
  x and y into a new set that is the union of these
  two sets.
• FIND-SET(x) Returns a pointer to the
  representative of the set containing x
• The number of union operations is atmost n-1.

18
MAKE-SET OPERATION

• Makes a singleton set
• Every set should have a name which should be any
  element of the set
• Make-Set(1)
• Make-Set(2)
• Make-Set(n)

19
UNION OPERATION

• Initially each number is a set by itself.
• From n singleton sets gradually merge to form
  a set.
• After n-1 union operations we get a single set
  of n numbers.

UNION Merge two sets and create a new set
3
1
4
2
20
FIND OPERATION

• Every set has a name.
• Thus Find(number) returns name of the set.
• It can be performed any number of times on the
  sets.
• The time taken for a find operation is O(n)
  whereas for Union operation it is O(1).

21
LINKED LIST REPRESENTATION

• Every Set is represented as linked list where the
  first element is the name of the set.
• We have the array of elements which have pointers
  to the elements in the linked lists.

5 2 1
3
4
7
3 is head of this set
3 4 7
5
1
2
5 is head if this set
10
10
10 is head of this singleton set
22
LINKED LIST REPRESENTATION

• For n Unions and m Finds, then time taken is
  nmn.
• If we have a pointer from each element in the set
  to the head, then the time to find operation is
  O(1).
• NOTE If m is large,
• Find O(nmn)
• Union O(1)

3 is the head
5 is the head
23
LINKED LIST REPRESENTATION
Each element pointing to the head i,e 3 in this
example
The 2 sets are being merged by connecting 5 and 7
In this case the union takes O(n2m) time.
24

• If we assume that the smaller set is attached to
  the end of the larger set in Union operation,
  then Union?O(n) and Find ?O(1)
• But in the AMORTIZED ANALYSIS, Average time taken
  for union is O(log n).
• So n Unions and m Finds take (nlog n m)
  time.
• To guarantee O(log n) time for Union, instead of
  pointing each element to the main head, point the
  Heads of the individual sets to a main head.

25

• If we consider path lengths to combine two
  trees(L1 is the path length of tree1 and L2 is
  the path length of tree2), then
• If L1gtL2 or L2gtL1, path length doesnt change
  i.e. It is still the longer path.
• If L1L2, then the path length is
• L21 if the head of tree1 is pointed to head of
  tree2
• L11 if the head of tree2 is pointed to head of
  tree1