data structures and algorithms 12 - edx · ming zhang “data structures and algorithms"...

Data Structures and Algorithms（12）

Instructor: Ming ZhangTextbook Authors: Ming Zhang, Tengjiao Wang and Haiyan Zhao

Higher Education Press, 2008.6 (the "Eleventh Five-Year" national planning textbook)

https://courses.edx.org/courses/PekingX/04830050x/2T2014/

Ming Zhang "Data Structures and Algorithms"

2

目录页

Ming Zhang “Data Structures and Algorithms"

Chapter 12

Advanced Data

Structure

Chapter 12 Advanced data structure

• 12.1 Multidimensional Array

• 12.2 Generalized Lists

• 12.3 Storage management

• 12.4 Trie

• 12.5 Improved binary search tree

– 12.5.1 Balanced binary search tree

• Concept and insertion operation of AVL tree

• Deletion operation and efficiency analysis of

AVL tree

– 12.5.2 Splay Tree

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Chapter 12

Advanced Data

Structure

Deletion in an AVL Tree

• Deletion is the reverse operation of

insertion. Deletion in an AVL tree is

almost the same with the deletion of

BST.

• Deletion in an AVL tree is a little bit

complicated.

12.5 Improved binary search tree

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Chapter 12

Advanced Data

Structure

Adjustment after Deletion

• Conditions of adjustment

– Height and balance factor has changed.

– Adjust these changes from the current node to the

root

• Balance factor needs to be modified

– Modify it

• If modification is not needed then stop the adjustment

– Use bool variable modified to mark, initialize it with

TRUE

– when modified = FALSE, stop the adjustment.


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Chapter 12

Advanced Data

Structure

Case 1

• Balance factor of the current node a is 0

– Its left or right subtree is cut, balance factor is 1 or -1

– modified = FALSE

– Modification will not influent the nodes above it. So

the adjustment is over.


0

hh

delete

1

hh-1

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Chapter 12

Advanced Data

Structure

Case 2

• Balance factor of the current node is not 0, but the taller

subtree is cut.

– Modify its balance factor to 0

– modified = TRUE

– Keep modifying upward


-1

hh+1

0

hhHeight

reduced

by1

delete

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Chapter 12

Advanced Data

Structure

Case 3.1

• Balance factor of the current node a is not zero. And the

shorter subtree is cut. So node a isn’t balanced.

• Assume the root of the taller subtree is b. There are three

cases.

• case 3.1:balance factor of b is zero

• Single rotation

• modified = FALSE


1

h

h

h

0

zag

1

hh-1

h

-1a

ba

b

delete

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Chapter 12

Advanced Data

Structure

Case 3.2

• Case 3.2: balance factor of b is equal to that of

a.

– Single rotation

– Balance factors of a and b change to zero

– modified =TRUE


1

h-1 h

1 0

h-1h-1

h

0

zagb

ba

a

h

delete

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Chapter 12

Advanced Data

Structure

Case 3.3

• Case 3.3: balance factor of b is opposite to that of a

– Double rotation, rotate b and c, then rotate c and a

– Because of balance factor of the new root is 0

– Other nodes need adjustment

– Besides modified = TRUE


1

h

h-1

-1

h-2

/h-1

Double

rotation

Height

reduced

by one

h-1 h-1

h-1 or h-2

0

h-1

/h-2

a

a

b

b

c

c

delete

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Chapter 12

Advanced Data

Structure

Series of adjustments after deletion• Series of adjustments

– Ancestor nodes may be unbalanced after the

adjustment

– Keep adjusting. These adjustments might

pass to the root.

• Find the grandfather of the node

deleted just now.

– Start single rotation or double rotation

– Rotation times is O(log n)


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Chapter 12

Advanced Data

Structure

Examples of deletion


-1

01

-10 0 0

0 00 0 -1 0

0

i

d m

b g k n

a c f h j l

e

-1

-11

-10 -1 0

00 0 -1 0

0

i

d l

b g k n

a c fh j

e

(a) Delete node m, replace m with node I (case 1)

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Chapter 12

Advanced Data

Structure



-1

-11

-10 -1 0

00 0 -1 0

0

i

d l

b g k n

a c f h j

e

(b) delete n (Case 3.2)

-1

-21

-10 -1

00 0 -1 0

0

i

dl

b g k

a c f h j

e

(c) Regard l as the root, doing LL

single rotation (Case 3.2)

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Chapter 12

Advanced Data

Structure



-2

0

1

-10

0

0

0 0 -1 0

0

i

d

lb g

k

a c

f h

j

e

(d) After LL single rotation, then

adjust its father node I, doing LR

double rotation (Case 3.3)

0

0

0

0

0

0 -1

0 0 0

g

k

d

h

l

b f

a cj

e

0

1

i

(e) Adjustment is done, AVL tree

become balanced

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Chapter 12

Advanced Data

Structure

Height of AVL tree

• Height of AVL tree with n nodes must be

O(log n)

• The maximum height of AVL tree with n

nodes is no more than K log2n

– K is a small factor

– AVL tree that is the most close to

unbalanced

– Construct a series of AVL tree T1, T

2, T

3,…


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Chapter 12

Advanced Data

Structure

T1

T3

T2

T4

Ti-1Ti-2

Ti

Ti ‘s height is i

Ti is the most unbalanced AVL tree. Deleting

any node will cause it to be unbalanced.

T1

Any other AVL

trees with height I

have more nodes

than Ti


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Chapter 12

Advanced Data

Structure

Proof of height

• We can see that:

t(1) = 2

t(2) = 4

t(i) = t(i-1) + t(i-2) + 1

(t(i)+1) = (t(i-1)+1) + (t(i-2)+1)

• As for i>2 these relations are very similar to

Fibonacci numbers:

F(0) = 0

F(1) = 1

F(i) = F(i-1) + F(i-2)


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Chapter 12

Advanced Data

Structure

Proof of height

• As for i>l, we have

t (i) = F (i+3) - 1

• Fibonacci number has this formula

• So the approximate formula is


2

51 here ，

5

1)(

iiF

15

1)i(t 3i

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Chapter 12

Advanced Data

Structure

Proof of height (result)

• Relation between height I and the number of nodes t(i)

• Use formula logφX = log2X / log

2φ and log

2φ ≈ 0.694 ,

calculate the approximate upper bound.

– t (i) = n

– So height of AVL tree with n nodes must be O (log n)


)1)i(t(53i

)1)i(t(log5log3i

2

3log (n 1) 1

2i

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Chapter 12

Advanced Data

Structure

Efficiency of AVL tree

• Time cost of searching, inserting and

deleting is O(1og2

n)

– Height of AVL tree with n nodes must be

O(log n)

• AVL tree is fit to data of small scale in

memory

• For large scale data stored in external

memory

– B tree or B+ tree


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Chapter 12

Advanced Data

Structure

Discussion

• Compare AVL tree with black-red tree,

which is better?

– Height of tree in the worst case

– Efficiency of operation in statistics

– Which one is easier to implement?


Data Structures and Algorithms

Thanks

the National Elaborate Course (Only available for IPs in China)http://www.jpk.pku.edu.cn/pkujpk/course/sjjg/

Ming Zhang, Tengjiao Wang and Haiyan ZhaoHigher Education Press, 2008.6 (awarded as the "Eleventh Five-Year" national planning textbook)

Ming Zhang “Data Structures and Algorithms”

data structures and algorithms 12 - edx · ming zhang “data structures and algorithms"...

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