Algorithms
Augmenting Data Structures:
Interval Trees
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Augmenting Data Structures - Introduction to Algorithms - Lecture Slides, Slides of Computer Science
These are the Lecture Slides of Introduction to Algorithms which includes Expensive Operations, Sort Edges, Running Time, Upshot, Union, Makeset, Disjoint Set, Disjoint Set Union, Naïve Implementation etc. Key important points are: Augmenting Data Structures, Interval Trees, Dynamic Order Statistics, Unordered, Support Finding, Dynamic Set, Statistic Operations, Trees Augment, Red Black Trees, Order Statistic Trees
Typology: Slides
2012/2013
1 / 37
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Algorithms
Augmenting Data Structures:
Interval Trees
Review: Dynamic Order Statistics
● We’ve seen algorithms for finding the i th
element of an unordered set in O( n ) time
● OS-Trees : a structure to support finding the i th
element of a dynamic set in O(lg n ) time
■ Support standard dynamic set operations
(Insert(), Delete(), Min(), Max(),
Succ(), Pred())
■ Also support these order statistic operations:
void OS-Select(root, i );
int OS-Rank( x );
Review: OS-Select
● Example: show OS-Select( root , 5):
M 8
C 5
P 2
Q 1
A 1
F 3
D 1
H 1
OS-Select(x, i)
{
r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r);
}
Review: OS-Select
● Example: show OS-Select( root , 5):
M 8
C 5
P 2
Q 1
A 1
F 3
D 1
H 1
OS-Select(x, i)
{
r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r);
}
i = 5 r = 6
Review: OS-Select
● Example: show OS-Select( root , 5):
M 8
C 5
P 2
Q 1
A 1
F 3
D 1
H 1
OS-Select(x, i)
{
r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r);
}
i = 5 r = 6
i = 5 r = 2
i = 3 r = 2
Review: OS-Select
● Example: show OS-Select( root , 5):
M 8
C 5
P 2
Q 1
A 1
F 3
D 1
H 1
OS-Select(x, i)
{
r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r);
}
i = 5 r = 6
i = 5 r = 2
i = 3 r = 2
i = 1 r = 1
Review: Determining The
Rank Of An Element
M
C
P
Q
A
F
D
H
OS-Rank(T, x)
{
r = x->left->size + 1;
y = x; while (y != T->root)
if (y == y->p->right)
r = r + y->p->left->size + 1;
y = y->p;
return r;
}
Idea: rank of right child x is one
more than its parent’s rank, plus
the size of x ’s left subtree
Review: Determining The
Rank Of An Element
M
C
P
Q
A
F
D
H
OS-Rank(T, x)
{
r = x->left->size + 1;
y = x; while (y != T->root)
if (y == y->p->right)
r = r + y->p->left->size + 1;
y = y->p;
return r;
}
Example 1:
find rank of element with key H
y
r = 1
Review: Determining The
Rank Of An Element
M
C
P
Q
A
F
D
H
OS-Rank(T, x)
{
r = x->left->size + 1;
y = x; while (y != T->root)
if (y == y->p->right)
r = r + y->p->left->size + 1;
y = y->p;
return r;
}
Example 1:
find rank of element with key H
r = 1
r = 3
y
r = 3+1+1 = 5
Review: Determining The
Rank Of An Element
M
C
P
Q
A
F
D
H
OS-Rank(T, x)
{
r = x->left->size + 1;
y = x; while (y != T->root)
if (y == y->p->right)
r = r + y->p->left->size + 1;
y = y->p;
return r;
}
Example 1:
find rank of element with key H
r = 1
r = 3
r = 5
y
r = 5
Review: Determining The
Rank Of An Element
M
C
P
Q
A
F
D
H
OS-Rank(T, x)
{
r = x->left->size + 1;
y = x; while (y != T->root)
if (y == y->p->right)
r = r + y->p->left->size + 1;
y = y->p;
return r;
}
Example 2:
find rank of element with key P
r = 1
y
r = 1 + 5 + 1 = 7
Review: Maintaining Subtree Sizes
● So by keeping subtree sizes, order statistic
operations can be done in O(lg n) time
● Next: maintain sizes during Insert() and
Delete() operations
■ Insert(): Increment size fields of nodes traversed
during search down the tree
■ Delete(): Decrement sizes along a path from the
deleted node to the root
■ Both: Update sizes correctly during rotations
Review: Methodology For
Augmenting Data Structures
● Choose underlying data structure
● Determine additional information to maintain
● Verify that information can be maintained for
operations that modify the structure
● Develop new operations
Interval Trees
● The problem: maintain a set of intervals
■ E.g., time intervals for a scheduling program:
7 10
5 11
4 8 15 18 21 23
17 19
i = [7,10]; i → low = 7; i → high = 10