lecture 16: zen & the art of o ( log n ) search

LECTURE 16:ZEN & THE ART OF O(LOG n) SEARCH

CSC 213 – Large Scale Programming

Today’s Goals

Review Map & Dictionary implementations What do they do well? When would they be

used? Why do they suck so much another is

needed? Discuss how BSTs can achieve Zen-like

balance Nodes will need more data, what fields do

we add? How this will modify approach to add &

remove data? What nodes get restructured and how to

know? Are there any neat hacks to coding

restructures? How traces help us write this tricky code

easily

Map & Dictionary ADT

Implementation

Searching

Adding Removing

Ordered List O(log n) O(n) O(n)Unordered List

O(n) O(n)/O(1) O(n)

Hash O(n) O(n) O(n) if lucky/good

O(1) O(1) O(1)

BST O(n) O(n) O(n) if balanced

O(log n) O(log n) O(log n)

Binary Search Trees

Maintain specific order Lower keys - left

subtree Right subtree for higher Equal keys not

specified, just be consistent

Fastest when complete Normally we see worst

case How to force this to

balance?

6

92

41 108

AVL Tree Definition

Fancy type of BST O(log n) time

provided For this, needs more

info

6

92

41 8

5

AVL Tree Definition



info

6

92

41 8

5

Node heights are shown in blue

1

21 1

23

4

AVL Tree Definition



info Keep tree balanced

by… Checking heights of

kids Only let differ by 0 or

1

6

92

41 8

5


1

1 1

3

4

2

2

AVL Tree Definition






1


6

92

41 8

51

1 1

3

4

2

2

AVL Tree Definition






1 Fix larger

differences by Shifting nodes in the BST

Help it maintain balance


6

92

41 8

51

1 1

3

4

2

2

AVL Tree Definition






1 Fix larger

differences by Shifting nodes in the BST

Help it maintain balance

Trinode Restructuring


6

92

41 8

51

1 1

3

4

2

2


Insertion & removal can unbalance tree Restore tree’s tao using trinode

restructuring When node’s children’s height differ

more than 1 Node, taller child, & tallest grandchild

used for this Grandchild used must be taller child of

taller child Median node subtree's root after

restructure Restructure makes the other 2 nodes its

children Also redistributes other child & grandkids


CASE 1: Single rotation (e.g., 3 in a row)

b

a

cT0

T1

T2 T3


CASE 1: Single rotation (e.g., 3 in a row) Move the taller child to subtree root Parent & grandchild become kids of the

taller child

b

a

cT0

T1

T2 T3

b

a c


CASE 1: Single rotation (e.g., 3 in a row) Move the taller child to subtree root Parent & grandchild become kids of the

taller child Does not (CANNOT) change order of nodes

in tree b

a

cT0

T1

T2 T3

b

a c

T0 T1 T2 T3


CASE 2: Double rotation (e.g., zig-zag)

T0

T1 T2

T3

b

a

c


CASE 2: Double rotation (e.g., zig-zag) Start when taller child & grandchild in

opposition Grandchild becomes root once this

completesc

a bT0

T1 T2

T3

b

a

c


CASE 2: Double rotation (e.g., zig-zag) Start when taller child & grandchild in

opposition Grandchild becomes root once this

completes Does not (CANNOT) change order of nodes

in tree

T0 T1 T2 T3

c

a bT0

T1 T2

T3

b

a

c

AVL Tree Insertion

Normal BST insertion starts the process Once added must then check tree for

balance insert(5)

1

7

92

41 81 1

6

2

3 2

4

AVL Tree Insertion


balance insert(5)

?

7

92

41 81 1

6?

?

? 2

?

5

Insertion in an AVL Tree

From inserted node, walk up to root At worst, stop after root has been

processed With each node, check if children are

balanced Restructure to restore balance among the

children Each insertion performs at most 1

restructuring Unbalanced node on path from root to

inserted node Most cases do not require a trinode

restructuring

AVL Tree Insertion


balance insert(5)

?

7

92

41 81 1

61

?

? 2

?

5

AVL Tree Insertion


balance insert(5)

2

7

92

41 81 1

61

?

? 2

?

5

AVL Tree Insertion


balance insert(5)

2

7

92

41 81 1

61

?

? 2

?

5

Oops

9


Uses the node, taller child, tallest grandchild Must be on insertion path, so used for

restructuring insert(5)

2

7

2

41 81 1

61

?

?

?

5

2

9

4


Uses the node, taller child, tallest grandchild Must be on insertion path, so used for

restructuring insert(5)

?

7

2

51 81 1

6?

?

? 2

?

9

4

After The Restructuring

Balance restored to tree; tao returns to normal Will need to complete walk updating

heights insert(5)

1

7

2

51 81 1

61

2

3 2

4

9

4

Removing From AVL Tree

Normal BST removal starts AVL tree removal Removal of node may also cause tao to

drop remove(9)

1

7

2

51 81 1

61

2

3 2

4



drop remove(9) 8

41

7

2

511

61

2

3 ?

?


Again walk up tree checking for balance Examine nodes along path only

There will be no change for nodes not on path

Multiple restructuring operations may be needed However, may not need any restructuring Only 1 restructuring could be needed

Use the node, taller child & tallest grandchild May not be on the path from start to root Works exactly like restructuring during

insert



drop remove(9) 8

41

7

2

511

61

2

3 ?

?



drop remove(9) 8

41

7

2

511

61

2

3 1

?


Uses the node, taller child, tallest grandchild Nodes may not be on path used during the

removal remove(9) 8

41

7

2

511

61

2

3 1

?

After the Restructure

Cannot stop till we reach the top of the tree May need to perform 0, 1, or multiple

restructures remove(9) 7

5

2

411 1

2 2

3

16

18

Restructuring for Dummies

Store the 7+1 nodes in local variables Record subtree’s parent in the “+1”

variable All of left, right, & parent set using

pattern Median node is root; subtrees maintain

order





order b

a

cT0

T1

T2 T3

b

a c

T0 T1 T2 T3





orderT0

T1 T2

T3

b

a

c

c

a b

T0 T1 T2 T3

For Next Lecture

Weekly assignment to demonstrate knowledge Due at regular time Tuesday

Program#1 tests due Monday at midnight Remember to submit updated design & test

cases Quiz on Monday on trees, BSTs, & AVL

trees Please study up on all the material covered Use your templates: they can greatly

simplify process Do not forget that pictures worth 1000

words

lecture 16: zen & the art of o ( log n ) search

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