bai 2 - advanced data structures

8/9/2019 Bai 2 - Advanced Data Structures

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Data Structures & Algorithms

Cc cu trc d liu nng cao

(Advanced data structures)

Nguyn Tri Tun

Khoa CNTT H.KHTN.Tp.HCMEmail: nttuan@ fit.hcmuns.edu.vn

Autumn 2008 Data Structures & A lgori thms -Advanced data structures -Nguyen Tri Tuan, DH.KHTN Tp.HCM 2

Advanced data structures

Review

Gii thiu

Cy en (Red Black Tree)

AA Tree


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Review

cao ca cy nh phn tm kim (BST) cnbng c N nodes l O(log2N)

Cy cn bng c chi ph thp

C nhiu cch xy dng cy nh phn tm kimcn bng:

AVL tree

Red-Black tree

AA tree

Splay tree


Gii thiu

Cc thut ng thng dng:

BST

AVL tree

Red Black tree

AA tree

Splay tree / Top-down splay tree


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Review

Gii thiu


AA Tree


Red Black Tree

nh ngha

Cu trc lu tr

Cc tnh cht

Cc thao tc c bn

nh gi


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Red Black Tree (tt)

nh ngha: Red-Black tree l mt cy nh phntm kim (BST) tun th cc quy tc sau:

[1] Mi node phi l hoc en

[2] Node gc l en

[3] Cc node ngoi (external node; NULL node) mcnh l nhng node en

[4] Nu mt node l , nhng node con ca n phil en

[5] Mi ng dn t gc n node ngoi phi c

cng s lng node en


Red Black Tree (tt)

Minh ha Red-Black tree

H = 4

Hb = 2

H = 3

Hb = 2

H = 2

Hb = 1

H = 1

Hb = 1

H = 1

Hb = 1

H = 3

Hb = 2

H = 2

Hb = 1


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Red Black Tree (tt)

Chiu cao en (black height hb(x)): l s nodeen trn ng i t node xn node ngoi(khng bao gm x)

T quy tc [4]khng th tn ti node cha vnode con cng . Khi cy en vi phm quitc ny gi l hin tng xung t -

Autumn 2008 Data Structures & A lgori thms -Advanced data structures -Nguyen Tr i Tuan, DH.KHTN Tp.HCM 10

Red Black Tree (tt)

Cu trc lu tr:Thng tin lu tr ti Node (key)

a chnode gc ca cy con bn tri (* pLeft)

a chnode gc ca cy con bn phi (* pRight)

a chca node cha (* pParent)

Thuc tnh mu ca node (color)


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Red Black Tree (tt)

typedef enum {BLACK, RED} NodeColor;

typedef int DataType; // Kiu d liu

typedef struct NodeTag {

DataType key; // D liu NodeColor color; // Mu ca node

struct NodeTag *pLeft;

struct NodeTag *pRight;

struct NodeTag *pParent; // d ci t} RBNode;

typedef struct RBNode* RBTREE;


Red Black Tree (tt)

Cc tnh cht:Tnh cht 1:

h: chiu cao ca cy

hb: chiu cao en

h


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Red Black Tree (tt)

Cc thao tc c bn:

Tm kim & duyt cy: ging BST. Do cy Red-Blackcn bng nn chi ph duyt cy tt hn BST

Thm node mi (insert node)

Xa node (delete node)


Red Black Tree (tt)

Insert node:

Thc hin ging nh cy BST

Node mi thm lun lun c mu

Nu xy ra vi phm qui tciu chnh cy

Demo chng trnh


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Red Black Tree (tt)

Insert node: (tt) nhng qui tc c th b vi phm

Mi node phi l hoc en OK

Node gc l en not OK ! Nu node mi l root

Cc node ngoi (NULL) phi lun lun en OK

Nu mt node l , nhng node con ca n phi l en notOK ! v c th parent[z] = RED 2 node lin tip mu

Mi ng dn t gc n nt l phi c cng s lng nodeen OK v khng lm thay i s node en


Red Black Tree (tt)

RB_Insert_Node(T, z) // T: cy; z: node mi

y NULL; x root[T];

while x NULL { // i n nt l

y x // y: node cha ca x

if (key[z] < key[x]) x left[x];

else x right[x];

}

parent[z] y; // thm node z vo cy

if (y == NULL) root[T] z; // l con ca node y

else if (key[z] < key[y]) left[y] z;

else right[y] z;

left[z] NULL

right[z] NULL

color[z] RED // node m i z c mu

RB_Insert_FixUp(T, z) // iu chnh cy


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Red Black Tree (tt)

Cch thc iu chnh cyPhp o mu

Php xoay tri (Left-Rotation)

Php xoay phi (Right-Rotation)


Red Black Tree (tt)

Php o mu

color[parent[z]] black

color[y] black

color[parent[parent[z]]] red

z = parent[parent[z]]

z

z

y


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Red Black Tree (tt)

Php xoay tri (Left-Rotation):


Red Black Tree (tt)

V d php xoay tri


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Red Black Tree (tt)

RB_Left_Rotate(T, x)

y right[x];

right[x] left[y];

if (left[y] NULL) parent[left[y]] x;

parent[y] parent[x];

if (parent[x] == NULL) root[T] y;

else if (x == left[parent[x]])

left[parent[x]] y;

else right[parent[x]] y;

left[y]x;

parent[x] y;


Red Black Tree (tt)

Php xoay phi (Right-Rotation):

RB_Right_Rotate(T, x): tng t hmxoay tri (t vit)


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Red Black Tree (tt)

Tng kt: c 6 trng hp x l chi titTrng hp 1: p dng php o mu

color[parent[z]] black

color[y] black

color[parent[parent[z]]] red

z = parent[parent[z]]


Red Black Tree (tt)

Tng kt: (tt)Trng hp 2: p dng php o mu v xoay phi

color[parent[z]] blackcolor[parent[parent[z]]] red

RIGHT-ROTATE(T, parent[parent[z]])


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Red Black Tree (tt)

Tng kt: (tt)Trng hp 3: p dng xoay tri a v dng 2

Case 3 Case 2

z parent[z]LEFT-ROTATE(T, z)

X l nh trng hp 2


Red Black Tree (tt)

11Thm 4

2 14

1 157

85

4

y

z

11

2 14

1 157

85

4

z

Case 1

y

Case 3

11

2

14

1

15

7

8

5

4

z

y Case 2

112

141

15

7

85

4

z


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Red Black Tree (tt)

Tng kt: (tt)3 trng hp tng t [1], [2], [3] i xng vi [1],[2], [3] qua trc Y

Case [1]

Case [2]

Case [3]


Red Black Tree (tt)

RB_Insert_FixUp(T, z)

while (color[parent[z]] == RED) {

// trng hp [1], [2], [3]

if (parent[z] == left[parent[parent[z]]]) {

y right[parent[parent[z]]];

if (color[y] == RED) Case 1;

else {

if (z == right[parent[z]]) Case 3;

Case 2;

}}

else // trng hp [1], [2], [3]

color[root[T]] BLACK


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Red Black Tree (tt)

nh gi thao tc Insert node:

Chi ph thm phn t mi (z): O(log2N)

Chi ph ca RB_Insert_FixUp: O(log2N)

Chi ph tng cng: O(log2N)


Red Black Tree (tt)

Delete node:Cch thc xa 1 node: ging nh BSTDemo chng trnhNu node b xo c mu : khng gy ra vi phm


Node gc l en OK

Cc node l (NULL) phi lun lun en OK

Nu mt node l , nhng node con ca n phi l enOK v khng to ra 2 node lin tip mu

Mi ng dn t gc n nt l phi c cng s lngnode en OK v khng lm thay i s node en


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Red Black Tree (tt)

Delete node: (tt)Nu node b xo c mu en: c th gy ra vi phm


Node gc l en not OK ! V c th xa root v thay bng node

Cc node l (NULL) phi lun lun en OK

Nu mt node l , nhng node con ca n phi l en not OK! V c th to ra 2 node lin tip mu

Mi ng dn t gc n nt l phi c cng s lng node en not OK ! V lm gim i s node en

Xem chi tit Data structure & Analysis in C, p. 465


Red Black Tree (tt)

nh gi:u im:

Chi ph tm kim O(log2N)

Insert O(log2N)

Delete O(log2N)

Minimum O(log2N)

Maximum O(log2N)

Hn ch:

Phi lu tr thuc tnh mu (color) v con trn nt cha(pParent)

Chi ph chn v xa cao hn BST v phi thc hin i muv php quay


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Review

Gii thiu


AA Tree


AA (Arne Andersson) Tree

Minh ha

Cc khi nim

nh ngha

Cu trc lu tr

Cc thao tc c bn

nh gi


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AA Tree (tt)

Minh ha cu trc cy AA

30 70

85

5

60

8010

90

15

20

50

35 40 6555

Lin kt

ngang

Lin kt

con tri

Lin kt

con phi


AA Tree (tt)

Cc khi nim:

Mc (Level) ca mt node

Lin kt ngang (Horizontal link)

Xoay phi (Right rotation Skew)

Xoay tri (Left rotation Split)


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AA Tree (tt)

Cc khi nim: (tt)Mc (Level) ca mt node: l s lin kt tri t node n NULL

Mc ca node NULL l 0

Mc ca node l l 1

BA

C ED

Mc 2

Mc 1


AA Tree (tt)

Cc khi nim: (tt)Lin kt ngang (Horizontal link): l lin kt gia mtnode v node con ca node cng mt mc

BA

C ED

Node cha

Node con

cng mc


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AA Tree (tt)

Cc khi nim: (tt)Xoay phi (Skew): xa b lin kt ngang bn tri

S dng iu chnh cy

X P

A B C

X P

A B C

tt1t


AA Tree (tt)

Cc khi nim: (tt)Xoay tri (Split): xo b 2 lin kt ngang lin tip bnphi

S dng iu chnh cy

X R

A B

GX

R

A B

G

t t1

t


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AA Tree (tt)

Skew c th to ra nhiu lin kt ngang phi lintip s dng Split iu chnh

5 103

5 103

5

103

Split

Skew


AA Tree (tt)

nh ngha: AA tree l mt cy nh phn tmkim (BST) tun th cc quy tc sau:

Lin kt ngang lun hng v bn phi

Khng c 2 lin kt ngang lin tip nhau

Mi node c mc > 1 s c 2 node con

Nu mt node khng c lin kt ngang phi th 2node con ca n cng mc


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AA Tree (tt)

30 70

85

5

60

8010

90

15

20

50

35 40 6555

Lin kt nganglun hng v

bn phiKhng c 2

lin kt ngang

lin tip

Node c

mc > 1

c 2 con

2 node con cng mc

khi node cha khng c

lin kt ngang


AA Tree (tt)

Tnh cht:Level ca node con tri lun nh hn level ca nodecha 1 n v

Level ca node con phi nh hn hay bng level canode cha (nhng khng qu 1 n v)

Thm mt node lun thc hin node c mc = 1


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AA Tree (tt)

Cu trc lu tr:

typedef int DataType; // Kiu d liu

typedef struct NodeTag {

DataType key; // D liu

struct NodeTag *pLeft;

struct NodeTag *pRight;

int level; // m c ca node

} AANode;

typedef struct AANode* AATREE;


AA Tree (tt)

Cc thao tc c bn:Khi thm 1 node

Node thm vo bn tri to ra mt lin kt ngang bn tri thc hin Skew

Node thm vo bn phi nu to ra 2 lin kt ngang lintip bn phi thc hin Split

30 70

85

5

60

8010

90

15

20

50

35 40 65553 48

Skew Split


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AA Tree (tt)

Cc thao tc c bn: (tt)Tm mt phn t: hon ton ging cy BST

30 70

85

5

60

8010

90

15

20

50

35 40 6555

Tm 55


AA Tree (tt)

Cc thao tc c bn: (tt)Thm mt node:

30 70

85

5

60

8010

90

15

20

50

35 40 655545

Thm 45Cn Split


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AA Tree (tt)


30 70

85

5

60

8010

90

15

20

50

35

40

655545

Sau khi Split ti 35

Cn Skew


AA Tree (tt)


30 70

85

5

60

8010

90

15

20

50

35

40

655545

Sau khi Skew ti 50

Cn Split


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AA Tree (tt)


Sau khi Split ti 40

Cn Skew

Cn Split

30 70

85

5

60

8010

90

15

20

50

35

40

655545


AA Tree (tt)


30 70

85

5

60

8010 90

15

20

50

35

40

655545

Sau khi Skew ti 70, v Split ti 30

STOP !


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AA Tree (tt)

AATREE AA_Insert_Node(DataType x, AATREE t){

if(t == NULL) { // to node mi v thm vo cyt = new AANode;

t->key = x;

t->pLeft = t->pRight = NULL;

t->level = 1;

}

else if(x < t->key)

t->pLeft = AA_Insert_Node(x, t->pLeft);

else if(x > t->key)

t->pRight = AA_Insert_Node(x, t->pRight);

else return t; // trng khat = Skew(t);

t = Split(t);

return t;

}


AA Tree (tt)AATREE right_rotate(AATREE &t)

{

AATREE t1;

t1 = t->pLeft;

t->pLeft = t1->pRight;

t1->pRight = t;

return t1;

}

AATREE Skew(AATREE &t)

{

if (t->pLeft != NULL)if (t->pLeft->level == t->level)

t = right_rotate(t);

return t;

}


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AA Tree (tt)

AATREE left_rotate(AATREE &t){

AATREE t1;

t1 = t->pRight;

t->pRight = t1->pLeft;

t1->pLeft = t;

t1->level++;

return t1;

}

AATREE Split(AATREE &t)

{

if (t->pRight !=NULL)

if (t->pRight->pRight != NULL)

if (t->pRight->pRight->level == t->level)

t = left_rotate(t);

return t;

}


AA Tree (tt)

Cc thao tc c bn: (tt)Thm mt node: VD. thm 6

10

131 113

2

4

8

5 7

12

9

6 ??


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AA Tree (tt)

Cc thao tc c bn: (tt)Xa mt node:

10

131 113

2

4

8

5 7

12

9

6

Xa 1

Gim mc


AA Tree (tt)


10

131132

4

8

5 7

12

9

6

Gim mc

Sau khi gim mc ti 2


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30


AA Tree (tt)


10

131132

4

8

5 7

12

9

6

Cn Skew

Sau khi gim mc ti 4 v 10


AA Tree (tt)


10

131132

4

8

5 7

12

9

6

Cn Skew

Sau khi Skew ti 10, ln 1


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AA Tree (tt)


10

131132

4 8

5 7

12

9

6

Cn Split

Sau khi Skew ti 10, ln 2


AA Tree (tt)


10

131132

4 8

5 7

12

9

6 Cn Split

Sau khi Split ti 4


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AA Tree (tt)


10

131132

4 8

5 7

12

9

6

Sau khi Split ti 8 STOP !


AA Tree (tt)

nh gi: phc tp O(log2N)

Khng cn lu con trn node cha (pParent)

Ci t n gin hn cy Red-Black


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bai 2 - advanced data structures

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