3. various data structure (2)

7/29/2019 3. Various Data Structure (2)

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B-TreesBinomial Heap

Fibonacci Heap


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B-Trees


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Motivation for B-Trees

Index structures for large datasets cannot be stored in main

memory.

Storing it on disk requires different approach to efficiency.

Assuming that a disk spins at 3600 RPM, one revolution

occurs in 1/60 of a second, or 16.7ms.

Crudely speaking, one disk access takes about the same time

as 200,000 instructions.


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Motivation (cont.)

Assume that we use an AVL tree to store about 20 million

records.

We end up with a very deep binary tree with lots of different

disk accesses; log2 20,000,000 is about 24, so this takes about0.2 seconds.

We know we cant improve on the log n lowerbound on

search for a binary tree.

But, the solution is to use more branches and thus reduce theheight of the tree!

As branching increases, depth decreases


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Definition of a B-tree

A B-tree of order m is an m-way tree (i.e., a tree where each

node may have up to m children) in which:

1. the number of keys in each non-leaf node is one less than the number

of its children and these keys partition the keys in the children in thefashion of a search tree.

2. all leaves are on the same level.

3. all non-leaf nodes except the root have at least m/ 2 children.

4. the root is either a leaf node, or it has from 2 to m children.

5. a leaf node contains no more than m1 keys.

The number m should always be odd


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An example B-Tree

51 6242

6 12

26

55 60 7064 9045

1 2 4 7 8 13 15 18 25

27 29 46 48 53

A B-tree oforder 5containing 26 items

Note that all the leaves are at the same level


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Inserting into a B-Tree

Attempt to insert the new key into a leaf:

If this would result in that leaf becoming too big, split the leaf into two,

promoting the middle key to the leafs parent.

If this would result in the parent becoming too big, split the parent into

two, promoting the middle key.

This strategy might have to be repeated all the way to the top.

If necessary, the root is split in two and the middle key is promoted to

a new root, making the tree one level higher.


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Suppose we start with an empty B-tree and keys arrive in thefollowing order:1 12 8 2 25 5 14 28 17 7 52 16 48 683 26 29 53 55 45

We want to construct a B-tree of order 5 The first four items go into the root:

To put the fifth item in the root would violate condition 5 Therefore, when 25 arrives, pick the middle key to make a

new root

Constructing a B-tree

1 2 8 12


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Constructing a B-tree (contd.)

1 2

8

12 25

6, 14, 28 get added to the leaf nodes:

1 2

8

12 146 25 28


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Adding 17 to the right leaf node would over-fill it, so we take the

middle key, promote it (to the root) and split the leaf

8 17

12 14 25 281 2 6

7, 52, 16, 48 get added to the leaf nodes

8 17

12 14 25 281 2 6 16 48 527


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Adding 68 causes us to split the right most leaf, promoting 48 to the root, and

adding 3 causes us to split the left most leaf, promoting 3 to the root; 26, 29, 53,

55 then go into the leaves

3 8 17 48

52 53 55 6825 26 28 291 2 6 7 12 14 16

Adding 45 causes a split of 25 26 28 29

and promoting 28 to the root then causes the root to split


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17

3 8 28 48

1 2 6 7 12 14 16 52 53 55 6825 26 29 45


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Exercise in Inserting a B-Tree

Insert the following keys to a 5-way B-tree:

3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56


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Removal from a B-tree

During insertion, the key always goes into a leaf. Fordeletion we wish to remove from a leaf. There are threepossible ways we can do this:

Case 1: - If the key is already in a leaf node, and removing itdoesnt cause that leaf node to have too few keys(m/ 2 -1),then simply remove the key to be deleted.

Case 2: - If the key is notin a leaf then it is guaranteed (by thenature of a B-tree) that its predecessor or successor will be ina leaf -- in this case we can delete the key and promote thepredecessor or successor key to the non-leaf deleted keys

position.


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Removal from a B-tree (2)

If (1) or (2) lead to a leaf node containing less than the

minimum number of keys then we have to look at the siblings

immediately adjacent to the leaf in question:

Case 3: if one of them has more than the min. number of keys then wecan promote one of its keys to the parent and take the parent key into

our lacking leaf

Case 4: if neither of them has more than the min. number of keys then

the lacking leaf and one of its neighbours can be combined with their

shared parent (the opposite of promoting a key) and the new leaf willhave the correct number of keys; if this step leave the parent with too

few keys then we repeat the process up to the root itself, if required


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Type #1: Simple leaf deletion

12 29 52

2 7 9 15 22 56 69 7231 43

Delete 2: Since there are enough

keys in the node, just delete it

Assuming a 5-way

B-Tree, as before...


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Type #2: Simple non-leaf deletion

12 29 52

7 9 15 22 56 69 7231 43

Delete 52

Borrow the predecessor

or (in this case) successor

56


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Type #4: Too few keys in node andits siblings

12 29 56

7 9 15 22 69 7231 43

Delete 72

Too few keys!

Join back together


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Type #4: Too few keys in node andits siblings

12 29

7 9 15 22 695631 43


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Type #3: Enough siblings

12 29

7 9 15 22 695631 43

Delete 22

Demote root key and

promote leaf key


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Type #3: Enough siblings

12

297 9 15

31

695643


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Exercise in Removal from a B-Tree

Given 5-way B-tree created by these data (last exercise):

3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56

Add these further keys: 2, 6,12

Delete these keys: 4, 5, 7, 3, 14


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Analysis of B-Trees

The maximum number of items in a B-tree of order m and height h:

root m1

level 1 m(m1)

level 2 m2(m1)

. . .

level h mh(m1)

So, the total number of items is

(1 + m + m2 + m3+ + mh)(m1) =

[(mh+11)/ (m1)] (m1) =mh+1

1

When m = 5 and h = 2 this gives 531 = 124


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Reasons for using B-Trees

When searching tables held on disc, the cost of each disctransfer is high but doesn't depend much on the amount ofdata transferred, especially if consecutive items are transferred

If we use a B-tree of order 101, say, we can transfer each node in onedisc read operation

A B-tree of order 101 and height 3 can hold 10141 items(approximately 100 million) and any item can be accessed with 3 discreads (assuming we hold the root in memory)

If we take m = 3, we get a 2-3 tree, in which non-leaf nodeshave two or three children (i.e., one or two keys)

B-Trees are always balanced (since the leaves are all at the samelevel), so 2-3 trees make a good type of balanced tree


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Comparing Trees

Binary trees

Can become unbalancedand lose their good time complexity (big O)

AVL trees are strict binary trees that overcome the balance problem

Heaps remain balanced but onlyprioritise (not order) the keys

Multi-way trees

B-Trees can be m-way, they can have any (odd) number of children

One B-Tree, the 2-3 (or 3-way) B-Tree, approximates a permanentlybalanced binary tree, exchanging the AVL trees balancing operations

for insertion and (more complex) deletion operations


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Binomial Heap


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Binomial Heaps

A way to implement mergeable heaps.

Useful in scheduling algorithms and graph algorithms.


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Definitions: Binomial Heap

Collection of binomial trees (satisfying some properties).


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Definitions: Binomial Trees

The binomial tree B(k) is an ordered tree defined recursively.

order of children is important.

1

32

1

23Different Trees

54 54

The binomial treeB(k) consists of two binomial treesB(k-1) that

are linkedtogether


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ExamplesB0

B1

B2

B3

B4

depth # nodes

0 1

1 4

2 6

3 4

4 1


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Another Way to Look at Bk

Bk-1

Bk-2B2

B1B0

Bk


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Properties of Binomial Trees

Properties of binomial tree Bk

it has 2knodes.

the height of the tree is k.

it has exactly Ckinodes at depth ifori= 0, 1,, k

Cki= n! /(i! (ni) !)

the root has degree k.


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Binomial Heaps - Definition

Binomial heap H is an ordered list of binomial trees that

satisfies the following properties:

Each binomial tree is heap-ordered, i.e. for each node its

key is greater or equal to the key of its parent.

There is at most one binomial tree in H whose root has

a given degree.

Binomial trees are contained in the list in the order of

increasing degrees.


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Binomial Heaps - Binomial Trees

B0 B2B1 B3


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B4


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Bk

Bk-1

Bk-2

B2B1

B0


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Binomial Heaps - Example

38

14 29

6

27

11 17

8

18

12 25

110


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Binomial Heaps - Union - Example

17

10 44

6

50

48 31

2937

318

24

23 22

8

55

45 32

30

25

712

33

15

41

28

H1 H2


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17

10 44

6

50

48 31

2937

318

24

23 22

8

55

45 32

30

25

712

33

15

41

28H1H2


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17

10 44

6

50

48 31

2937

3

18

24

23 22

8

55

45 32

30

25

712

33

15

41

28

H1H2


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17

10 44

6

50

48 31

2937

3

18

24

23 22

8

55

45 32

3025

7

12

33

15

41

28

H1H2


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17

10 44

6

50

48 31

2937

3

18

24

23 22

8

55

45 32

3025

7

12

33

15

41

28

H1H2

A l


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Analogy

37

3

50

31 1748

441029

6

55

32 2445

222330

8

33

41

28

15

25

7

head[H]

18

12

prev-x x

12

33

41

28

15

25

7head[H1] 18

37

3

50

31 1748

441029

6

55

32 2445

222330

8

head[H2]

Union

Like binary addition:

1 1 1(carries)

0 0 1 1 1

1 0 0 1 1

1 1 0 1 0

temporarily have three

trees of this degree

Bi i l H E Mi


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Binomial Heaps - ExtractMin -Example

17

10 44

6

50

48 31

2918

24

23 22

8

55

45 32

30

12

33

15

41

28

Bi i l H E t tMi


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17

10 44

6

50

48 31

2918

24

23 22

8

55

45 32

30

12

33

15

41

28

Bi i l H E t tMi


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17

1044

50

48 31

29

18

24

23 22

8

55

45 32

30

12

33

15

41

28

Bi i l H E t tMi


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17

1044

50

48 31

29

18

24

23 22

8

55

45 32

30

12

33

15

41

28


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Binomial Heaps - ExtractMin -

Example

17

1044

50

48 31

29

18 24

23 22

8

55

45 32

3012 33

15

41

28


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Example

17

1044

50

48 31

29

18 24

23 22

8

55

45 32

3012 33

15

41

28


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Example

17

1044

50

48 31

2918 24

23 22

8

55

45 32

3012

33

15

41

28

Bi i l H D K


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Binomial Heaps - DecreaseKey -Example

17

1044

50

48 31

2918 24

23 22

8

55

45 32

3012

33

15

41

28

10

Binomial Heaps DecreaseKe


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17

1044

10

48 31

2918 24

23 22

8

55

45 32

3012

33

15

41

28

Binomial Heaps DecreaseKey


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17

1044

48

10 31

2918 24

23 22

8

55

45 32

3012

33

15

41

28



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17

1044

48

29 31

1018 24

23 22

8

55

45 32

3012

33

15

41

28



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17

1044

48

29 31

1518 24

23 22

8

55

45 32

3012

33

10

41

28


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Binomial Heaps - Summary

Binomial Heaps Heaps

Min (log n) or(1) (1)

ExtractMin (log n) (log n)

DecreaseKey (log n) (log n)

Union (log n) (n)

Insert (log n) (log n)

Delete (log n) (log n)

MakeEmpty (1) (1)

IsEmpty (1) (1)

3. various data structure (2)

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