Data Structures and Algorithms

Discussion 4: Week of Sep 26, 2011

Outline(

•  Heaps(•  Huffman(Encoding(•  AVL(Trees(

Heap%Review�•  Heapify(

;(PercolateUp,(TrickleDown(;(O(logN)(Eme(for(both(cases(

•  BuildHeap(;(O(N)(Eme,(proof(is(non;trivial(

•  DequeueMax(;(Replace(root(with(last(node,(then(fix(down(;(O(logN)(Eme(

Max;Heapify(2)(

heap-size[A] = 10�

Building%a%heap((

Build;Max;Heap(A)(1(((heap;size[A](←(length[A](2%%%for(i(←(!length[A]/2"((downto(1(3(( (((((do(Max;Heapify(A,(i)(

Chapter 6 P.5

Building%a%heap�

Building%a%heap� Building%a%heap�

DequeueMax�21

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14 10

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2

Delete(21.(

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14 10

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2

Move(2(to(root.(

2

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14 10

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Not(max(heap!(

Trickle(down!(

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14 10

2 Max(heap!(

DequeueMax�

20

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14 10

2

Delete(20.(

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14 10

2 Move(10(to(root.(

10

15

14

2

Not(max(heap!(

Trickle(down!(

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10

14

2

DequeueMax�

Not(max(heap!(

Trickle(down!(

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10

2

Max(heap!(

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10

14

2

Huffman(Encoding(

•  Compresses(data(to(use(less(space.(•  Works(best(on(text(data,(not(binary(data.(

– Why?(

Huffman(Encoding(

•  Compresses(data(to(use(less(space.(•  Works(best(on(text(data,(not(binary(data.(

– Huffman(encoding(uses(a(character’s(frequency(in(order(to(encode(things(–(the(more(frequent(a(character,(the(be_er(the(compression.(

– Binary(data(looks(like:(páW.ÈDŽOŸuæ`ç\lß(– Text(data(is(relaEvely(uniform(and(predictable.(

Huffman(Encoding�

Given(the(text(ALIBABA(which(of(the(following(can(be(the(Huffman(prefix(code(for(compressing(this(text?(A.(A=001,(B=000,(I=01,(L=10.(B.(A=00,(B=01,(I=10,(L=11.(C.(A=0,(B=11,(I=001,(L=000.(D.(A=0,(B=01,(I=100,(L=110.(E.(A=1,(B=01,(I=001,(L=000.(

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=001,(B=000,(I=01,(L=10.((

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=001,(B=000,(I=01,(L=10.(No.((

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=00,(B=01,(I=10,(L=11.(((

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=00,(B=01,(I=10,(L=11.(No.((

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=0,(B=11,(I=001,(L=000.((

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=0,(B=11,(I=001,(L=000.(Yes.((

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=0,(B=01,(I=100,(L=110(

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=0,(B=01,(I=100,(L=110.(Yes.(

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=1,(B=01,(I=001,(L=000.(

Huffman(Encoding(

•  ALIBABA(– A:(3(– B:(2(– L(:(1(–  I(:(1(

•  A=1,(B=01,(I=001,(L=000.(Yes.(

AVL Trees�

•  A self-balancing binary search tree •  Balance Factor

- Height(left-sub-tree) – Height(right-sub-tree) •  Balance factor of -1, 0, or 1 for each node •  Rotations needed for maintain balance

- Single Rotation (LL or RR) - Double Rotation(LR or RL)�

Single Rotation�

Double Rotation�Example

! Is this AVL balanced?

! Why/why not? 14

10

6

18

Example ! What happens

when you add (one after the other): !  4 !  12 !  20

14

10

6

18

AVL Trees ! Adding 4

14

10

6

18

Example ! Adding 4

14

10

6

18

4

Example ! Adding 4

!  Not AVL balanced !  What do we do?

14

10

6

18

4

Example ! Adding 4

!  Not AVL balanced !  What do we do? !  Right rotation

around 10

14

10

6

18

4

Example ! Adding 4

!  Not AVL balanced !  What do we do? !  Right rotation

around 10

14

6

4

18

10

Example ! Adding 12

14

6

4

18

10

Example ! Adding 12

14

6

4

18

10

12

Example ! Adding 12

!  Not AVL balanced !  What do we do?

14

6

4

18

10

12

Example ! Adding 12

!  Not AVL balanced !  What do we do? !  Double rotation

14

6

4

18

10

12

Example ! Adding 12

!  Not AVL balanced !  What do we do? !  Double rotation !  Left rotation

around 6

14

6

4

18

10

12

Example ! Adding 12

!  Not AVL balanced !  What do we do? !  Double rotation !  Left rotation

around 6

14

10

6

18

12

4

Example ! Adding 12

!  Not AVL balanced !  What do we do? !  Double rotation !  Left rotation

around 6 !  Right rotation

around 14

14

10

6

18

12

4

Example ! Adding 12

!  Not AVL balanced !  What do we do? !  Double rotation !  Left rotation

around 6 !  Right rotation

around 14

10

6

4

14

12 18

Example ! Adding 20

10

6

4

14

12 18

Example ! Adding 20

10

6

4

14

12 18

20

Example ! Adding 20

!  Still AVL Balanced 10

6

4

14

12 18

20