CS 332: Algorithms
Dynamic Order Statistics
David Luebke
1
7/1/2016
OS-Select Example
● Example: show OS-Select(root, 5):
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);
}
David Luebke
2
M
8
C
5
A
1
P
2
F
3
D
1
Q
1
H
1
7/1/2016
OS-Select Example
● Example: show OS-Select(root, 5):
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);
}
David Luebke
3
M
8
i=5
r=6
C
5
A
1
P
2
F
3
D
1
Q
1
H
1
7/1/2016
OS-Select Example
● Example: show OS-Select(root, 5):
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);
}
David Luebke
4
M
8
C
5
A
1
i=5
r=6
i=5
r=2
P
2
F
3
D
1
Q
1
H
1
7/1/2016
OS-Select Example
● Example: show OS-Select(root, 5):
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);
}
David Luebke
5
M
8
C
5
A
1
i=5
r=2
F
3
D
1
i=5
r=6
P
2
i=3
r=2
Q
1
H
1
7/1/2016
OS-Select Example
● Example: show OS-Select(root, 5):
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);
}
David Luebke
6
M
8
C
5
A
1
P
2
i=5
r=2
F
3
D
1
i=5
r=6
i=3
r=2
H
1
Q
1
i=1
r=1
7/1/2016
OS-Select: A Subtlety
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);
}
Oops…
● What happens at the leaves?
● How can we deal elegantly with this?
David Luebke
7
7/1/2016
OS-Select
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);
}
● What will be the running time?
David Luebke
8
7/1/2016
Determining The
Rank Of An Element
M
8
C
5
P
2
A
1
F
3
Q
1
D
1
H
1
What is the rank of this element?
David Luebke
9
7/1/2016
Determining The
Rank Of An Element
M
8
C
5
P
2
A
1
F
3
Q
1
D
1
H
1
Of this one? Why?
David Luebke
10
7/1/2016
Determining The
Rank Of An Element
M
8
C
5
P
2
A
1
F
3
Q
1
D
1
H
1
Of the root? What’s the pattern here?
David Luebke
11
7/1/2016
Determining The
Rank Of An Element
M
8
C
5
P
2
A
1
F
3
Q
1
D
1
H
1
What about the rank of this element?
David Luebke
12
7/1/2016
Determining The
Rank Of An Element
M
8
C
5
P
2
A
1
F
3
Q
1
D
1
H
1
This one? What’s the pattern here?
David Luebke
13
7/1/2016
OS-Rank
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;
}
● What will be the running time?
David Luebke
14
7/1/2016
OS-Trees: Maintaining Sizes
● So we’ve shown that with subtree sizes, order
statistic operations can be done in O(lg n) time
● Next step: maintain sizes during Insert() and
Delete() operations
■ How would we adjust the size fields during
insertion on a plain binary search tree?
David Luebke
15
7/1/2016
OS-Trees: Maintaining Sizes
● So we’ve shown that with subtree sizes, order
statistic operations can be done in O(lg n) time
● Next step: maintain sizes during Insert() and
Delete() operations
■ How would we adjust the size fields during
insertion on a plain binary search tree?
■ A: increment sizes of nodes traversed during search
David Luebke
16
7/1/2016
OS-Trees: Maintaining Sizes
● So we’ve shown that with subtree sizes, order
statistic operations can be done in O(lg n) time
● Next step: maintain sizes during Insert() and
Delete() operations
■ How would we adjust the size fields during
insertion on a plain binary search tree?
■ A: increment sizes of nodes traversed during search
■ Why won’t this work on red-black trees?
David Luebke
17
7/1/2016
Maintaining Size Through Rotation
y
19
x
11
6
x
19
rightRotate(y)
7
leftRotate(x)
4
y
12
6
4
7
● Salient point: rotation invalidates only x and y
● Can recalculate their sizes in constant time
■ Why?
David Luebke
18
7/1/2016
Augmenting Data Structures:
Methodology
●
Choose underlying data structure
■
●
Determine additional information to maintain
■
●
E.g., subtree sizes
Verify that information can be maintained for
operations that modify the structure
■
●
E.g., red-black trees
E.g., Insert(), Delete()
(don’t forget rotations!)
Develop new operations
■
David Luebke
E.g., OS-Rank(), OS-Select()
19
7/1/2016