[PPT] - Data Structures Splay Tree & Graph Virendra Singh Associate PowerPoint Presentation

SLIDE 1

Data Structures

Splay Tree & Graph

Virendra Singh

Associate Professor Computer Architecture and Dependable Systems Lab Department of Electrical Engineering Indian Institute of Technology Bombay

http://www.ee.iitb.ac.in/~viren/ E-mail: viren@ee.iitb.ac.in

EE-717/453:Advance Computing for Electrical Engineers

Lecture 10 (22 Aug 2013)

SLIDE 2

Splay Trees

Binary search trees.
Search, insert, delete, and split have amortized

complexity O(log n) & actual complexity O(n).

Two varieties.

– Bottom up. – Top down.

22 Aug 2013 EE-717/EE-453@IITB 2

SLIDE 3

Bottom-Up Splay Trees

Search, insert, delete, and join are done as in an

unbalanced binary search tree.

Search, insert, and delete are followed by a splay
peration that begins at a splay node.
When the splay operation completes, the splay

node has become the tree root.

Join requires no splay (or, a null splay is done).
For the split operation, the splay is done in the

middle (rather than end) of the operation.

22 Aug 2013 EE-717/EE-453@IITB 3

SLIDE 4

Splay Node – search(k)

If there is a pair whose key is k, the node containing

this pair is the splay node.

Otherwise, the parent of the external node where

the search terminates is the splay node.

20 10 6 2 8 15 40 30 25

22 Aug 2013 EE-717/EE-453@IITB 4

SLIDE 5

Splay Node – insert(newPair)

If there is already a pair whose key is newPair.key,

the node containing this pair is the splay node.

Otherwise, the newly inserted node is the splay

node.

2 1 6 2 8 1 5 4 3 2 5

22 Aug 2013 EE-717/EE-453@IITB 5

SLIDE 6

Splay Node – delete(k)

If there is a pair whose key is k, the parent of the

node that is physically deleted from the tree is the splay node.

Otherwise, the parent of the external node where

the search terminates is the splay node.

20 10 6 2 8 15 40 30 25

22 Aug 2013 EE-717/EE-453@IITB 6

SLIDE 7

Splay

Let q be the splay node.
q is moved up the tree using a series of splay

steps.

In a splay step, the node q moves up the tree by

0, 1, or 2 levels.

Every splay step, except possibly the last one,

moves q two levels up.

22 Aug 2013 EE-717/EE-453@IITB 8

SLIDE 8

Splay Step

If q = null or q is the root, do nothing

(splay is over).

If q is at level 2, do a one-level move

and terminate the splay operation.

p q a b c b c a q p

q right child of p is symmetric.

22 Aug 2013 EE-717/EE-453@IITB 9

SLIDE 9

Splay Step

If q is at a level > 2, do a two-level

move and continue the splay operation.

q right child of right child of gp is symmetric.

p q a b c gp d c d b p gp q a

22 Aug 2013 EE-717/EE-453@IITB 10

SLIDE 10

2-Level Move (case 2)

q left child of right child of gp is symmetric.

p q b c a gp d a c b gp p q d

22 Aug 2013 EE-717/EE-453@IITB 11

SLIDE 11

Per Operation Actual Complexity

Start with an empty splay tree and

insert pairs with keys 1, 2, 3, …, in this

rder.

1 1 2 1 2

22 Aug 2013 EE-717/EE-453@IITB 12

SLIDE 12

Per Operation Actual Complexity

Start with an empty splay tree and

insert pairs with keys 1, 2, 3, …, in this

rder.

1 2 3 1 2 3 1 2 3 4

22 Aug 2013 EE-717/EE-453@IITB 13

SLIDE 13

Per Operation Actual Complexity

Worst-case height = n.
Actual complexity of search, insert,

delete, and split is O(n).

22 Aug 2013 EE-717/EE-453@IITB 14

SLIDE 14

Binary Trie

Information Retrieval.
At most one key comparison per
peration.
Fixed length keys.

– Branch nodes.

Left and right child pointers.
No data field(s).

– Element nodes.

No child pointers.
Data field to hold dictionary pair.

22 Aug 2013 EE-717/EE-453@IITB 15

SLIDE 15

Example

At most one key comparison for a search.

0001 0011 1100 1000 1001 1 1 1 1

22 Aug 2013 EE-717/EE-453@IITB 16

SLIDE 16

Variable Key Length

Left and right child fields.
Left and right pair fields.

– Left pair is pair whose key terminates at

root of left subtree or the single pair that might otherwise be in the left subtree.

– Right pair is pair whose key terminates at

root of right subtree or the single pair that might otherwise be in the right subtree.

– Field is null otherwise.

22 Aug 2013 EE-717/EE-453@IITB 17

SLIDE 17

Example

At most one key comparison for a search.

1 null 00 01100 0000 001 10 11111 00100 001100 1000 101 1

22 Aug 2013 EE-717/EE-453@IITB 18

SLIDE 18

Fixed Length Insert

Insert 0111.

0001 0011 1100 1000 1001 1 1 1 1 0111 1

Zero compares.

22 Aug 2013 EE-717/EE-453@IITB 19

SLIDE 19

Fixed Length Insert

Insert 1101.

0001 0011 1100 1000 1001 1 1 1 1 0111 1

22 Aug 2013 EE-717/EE-453@IITB 20

SLIDE 20

Fixed Length Insert

Insert 1101.

1100 1 0001 0011 1000 1001 1 1 1 0111 1

22 Aug 2013 EE-717/EE-453@IITB 21

SLIDE 21

Fixed Length Insert

Insert 1101.

1 0001 0011 1000 1001 1 1 1 0111 1

One compare.

1100 1101 1

22 Aug 2013 EE-717/EE-453@IITB 22

SLIDE 22

Fixed Length Delete

Delete 0111.

1 0001 0011 1000 1001 1 1 1 0111 1 1100 1101 1

22 Aug 2013 EE-717/EE-453@IITB 23

SLIDE 23

Fixed Length Delete

Delete 0111. One compare.

1 0001 0011 1000 1001 1 1 1 1100 1101 1

22 Aug 2013 EE-717/EE-453@IITB 24

SLIDE 24

Fixed Length Delete

Delete 1100.

1 0001 0011 1000 1001 1 1 1 1100 1101 1

22 Aug 2013 EE-717/EE-453@IITB 25

SLIDE 25

Fixed Length Delete

Delete 1100.

1 0001 0011 1000 1001 1 1 1 1101 1

22 Aug 2013 EE-717/EE-453@IITB 26

SLIDE 26

Fixed Length Delete

Delete 1100.

1 0001 0011 1000 1001 1 1 1 1101

22 Aug 2013 EE-717/EE-453@IITB 27

SLIDE 27

Fixed Length Delete

Delete 1100.

1 0001 0011 1000 1001 1 1 1 1101

22 Aug 2013 EE-717/EE-453@IITB 28

SLIDE 28

Fixed Length Delete

Delete 1100. One compare.

1 0001 0011 1000 1001 1 1 1 1101

22 Aug 2013 EE-717/EE-453@IITB 29

SLIDE 29

Compressed Binary Tries

No branch node whose degree is 1.
Add a bit# field to each branch node.
bit# tells you which bit of the key to use to

decide whether to move to the left or right subtrie.

22 Aug 2013 EE-717/EE-453@IITB 30

SLIDE 30

Binary Trie

1 0001 0011 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4

bit# field shown in black outside branch node.

22 Aug 2013 EE-717/EE-453@IITB 31

SLIDE 31

Compressed Binary Trie

1 0001 0011 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4

bit# field shown in black outside branch node.

22 Aug 2013 EE-717/EE-453@IITB 32

SLIDE 32

Compressed Binary Trie

1 0001 0011 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4

#branch nodes = n – 1.

22 Aug 2013 EE-717/EE-453@IITB 33

SLIDE 33

Insert

1 0001 0011 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4

Insert 0010.

22 Aug 2013 EE-717/EE-453@IITB 34

SLIDE 34

Insert

Insert 0100.

1 0001 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4 0010 0011 1 4

22 Aug 2013 EE-717/EE-453@IITB 35

SLIDE 35

Insert

1 0001 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4 0010 0011 1 4 2 0100 1

22 Aug 2013 EE-717/EE-453@IITB 36

SLIDE 36

Delete

0001 1 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4 0010 0011 1 4 2 0100 1

Delete 0010.

22 Aug 2013 EE-717/EE-453@IITB 37

SLIDE 37

Delete

0001 1 1000 1001 1 1 1 1100 1101 1 1 2 3 4 4 0011 2 0100 1

Delete 1001.

22 Aug 2013 EE-717/EE-453@IITB 38

SLIDE 38

Delete

0001 1 1000 1 1 1100 1101 1 1 2 3 4 0011 2 0100 1

22 Aug 2013 EE-717/EE-453@IITB 39

SLIDE 39

Graphs

G = (V,E)
V is the vertex set.
Vertices are also called nodes and

points.

E is the edge set.
Each edge connects two different

vertices.

Edges are also called arcs and lines.
Directed edge has an orientation (u,v).

u v

22 Aug 2013 EE-717/EE-453@IITB 40