Design and Analysis of Algorithms(CS345/CS345A)

Lecture 6

Augmented BST for • Dynamic sequences

• Orthogonal range searching

1

The fundamental question we answered in last class

Question: What makes BST pervasive in the world of data structures ?

Answer: Augmentation : “storing extra fields at each node”

: Additional information

DATA STRUCTURES FORDYNAMIC SEQUENCES

3

Dynamic Sequence

4

OR

OR

Representing sequence using a BST

8

4

1 2

1

3

1 1

: size field

6

u

7

T

We need to do the rebalancing of

height later on.

Example 2: sequence of numbers

8

Performing Add(T,i,j,x)

9

iT

View the entire tree T from perspective of the paths from i

and j to the root. How will T look like ?

j

LCA(i,j)

Performing Add(T,i,j,x)

10

T

j

i

LCA(i,j)

What are the nodes whose values has to be incremented by x ?

Performing Add(T,i,j,x)

11

T

What are the nodes whose values has to be incremented by x ?

j

i

LCA(i,j)

Performing Add(T,i,j,x)

12

T

What are the nodes whose values has to be incremented by x ?

j

i

LCA(i,j)

Performing Add(T,i,j,x)

13

Key observations about the data structure

14

Homework

15

Example 2: sequence of bits

16

Example 3: sequence of numbers

17

ORTHOGONAL RANGE SEARCHING

Hopefully you have got a fair amount of understanding of augmenting a BST by now. Think over this problem for a while before proceeding

further.

18

Orthogonal Range searching

19

Rectangle

No. of points in query rectangle

Orthogonal Range searching

20

Rectangle

Orthogonal Range searching

21

22

T

LCA

• Binary search tree on x-coordinates

23

T

LCA

• Binary search tree on x-coordinates

24

T

LCA

• Binary search tree on x-coordinates

25

T

LCA

How should we augment each node?

26

left right

valueColor-bit

Augment each node v with a pointer to another BSTstoring all points of T(v) according to y-oordinates.

Y-tree

v

BST storing T(v)

according to y-coordinate

27

T

LCA

Homework

28

Space of the data structure

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T

1

2

3

4

. . .. . .

BST before augmentation

Space of the data structure

30

T

. . .

. . .

BST before augmentation

Total time to build the data structure

31

p