[Slide] Chuong 1 - Cac thuat toan sap xep.pdf

Cc thut ton sp xp(Sorting algorithms)

Nguyn Tri TunKhoa CNTT H.KHTN.Tp.HCMEmail: [email protected]

Spring 2009 Data Structure & Algorithm - Nguyen Tri Tuan - Khoa CNTT H KHTN Tp.HCM 2

Sp xp 1 mng cc s nguynSp xp 1 mng cc s nguyn

Gi s c 1 mng gm 6 s nguyn. Ta cn sp xp cc phn t ca mng theo th t tng dn

[0] [1] [2] [3] [4] [5]


Thut ton Chn trc tip (Selection sort Algorithm)Thut ton Chn trc tip (Selection sort Algorithm) Bt u bng

cch tm phn t nh nht

[0] [1] [2] [3] [4] [5]


Selection sort AlgorithmSelection sort Algorithm

Hon v phn t nh nht tm c vi phn t u tin ca mng

[0] [1] [2] [3] [4] [5]



1 phn ca mng c sp xp

Phn sp Phn cha sp

[0] [1] [2] [3] [4] [5]



Tm phn t nh nhttrong phn cha c sp

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Hon v phn t nh nht trong phn cha c sp vi phn t u tin trong phn ny [0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Phn c sp xp ca mng c tng thm 1 phn t

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Tip tc tng t ... Phn t

nh nht

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Tip tc tng t ...

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Tip tc tng t ...

Phn sp tng thm

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Qu trnh ln lt thm tng phn t vo phn sp

Phn sp cha cc phn t nh nht, sp tng dn

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Thut ton dng khi ch cn 1 phn t( l phn t ln nht).

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha



Ton b mng c sp th t.

Tng qut: chnphn t nh nht v a n v v tr u ca phn cha c sp trong mng.

[0] [1] [2] [3] [4] [5]


Selection sort Algorithm(Minh ha chng trnh)Selection sort Algorithm(Minh ha chng trnh)

void SelectionSort (int a[ ], int n ) {

int min; // v tr ca phn t nh nht (trong phn cha sp)int tmp; // bin tm dng khi hon vfor (int i = 0; i < n; i++ ) {

// tm phn t nh nht trong phn cha spmin = i;for (int j = i + 1; j < n; j++)

if (a[j] < a[min] ) min = j;// hon v phn t nh nht c tm thy vi phn t uif (a[min] < a[i]) { tmp = a[i]; a[i] = a[min]; a[min] = tmp; }

} // end of for i}


nh gi thut ton(Selection sort Algorithm)nh gi thut ton(Selection sort Algorithm)Trong mi trng hp, s php so snh l:

(n-1) + (n-2) + + 1 = n(n-1)/2 = O(n2)

S php hon v: Trng hp xu nht: O(n)Trng hp tt nht (mng sp t t tng

dn): 0


Thut ton Chn trc tip (Insertion sort Algorithm)Thut ton Chn trc tip (Insertion sort Algorithm) Thut ton

Chn trc tip cng chia mng thnh 2 phn: phn c spv phn cha c sp [0] [1] [2] [3] [4] [5]


Insertion sort AlgorithmInsertion sort Algorithm

Phn sp lc u ch c 1 phn t u tinca mng (khng cn thit l phn t nh nht)

Phn sp Phn cha sp

[0] [1] [2] [3] [4] [5]



M rng phn sp bng cch thm vo phn t u tin trong phn cha c sp

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



...v t n vo v tr thch hp, sao cho phn sp vn gi nguyn tnh th t (tng dn).

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Trong v d ny, phn t mi c t vo v tr u ca phn sp.

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



i khi chng ta gp may, phn t mi khng cn phi di chuyn.

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



v li gp may thm 1 ln na..

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp


Insertion sort AlgorithmLm sao chn 1 phn t ?Insertion sort AlgorithmLm sao chn 1 phn t ?

Copy phn t mi vo 1 bin trung gian

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha sp



Dch chuyn cc phn t trong phn sp sang phi

[0] [1] [2] [3] [4] [5]



to 1 ch trng cho phn t mi

[0] [1] [2] [3] [4] [5]



tip tc dch chuyn cc phn t...

[0] [1] [2] [3] [4] [5]



...cho n khi tm thy v tr thch hp cho phn t mi...

[0] [1] [2] [3] [4] [5]



Copy phn t mi vo li mng, ti v tr ny.

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha



Phn t cui cng cng phi chn. Copy n vo 1 bin trung gian...

[0] [1] [2] [3] [4] [5]

Phn sp Phn cha


Insertion sort AlgorithmCu hi ?Insertion sort AlgorithmCu hi ?

C bao nhiu php dch chuyn xy ra ?

[0] [1] [2] [3] [4] [5]


Insertion sort AlgorithmCu hi ?Insertion sort AlgorithmCu hi ?

C 4 php dch chuyn

[0] [1] [2] [3] [4] [5]



Copy phn t tr li mng.

[0] [1] [2] [3] [4] [5]


Insertion sort Algorithm(Minh ha chng trnh)Insertion sort Algorithm(Minh ha chng trnh)

void InsertionSort (int a[ ], int n) {

int saved; // bin trung gian lu li gi tr phn t cn chnfor (int i = 1; i < n; i++ ) {

saved = a[i]; // lu li gi tr phn t cn chn// dch chuyn cc phn t trong phn sp sang phifor (int j = i; j > 0 && saved < a[j-1]; j--)

a[j] = a[j-1];a[j] = saved; // chn phn t vo ng v tr

} // end of for i}


nh gi thut ton(Insertion sort Algorithm)nh gi thut ton(Insertion sort Algorithm)Trng hp xu nht c:

1 + 2 + 3 + + (n-1) = n(n-1)/2 = O(n2)php so snh v dch chuyn

Trng hp tt nht (mng c th t tng dn): O(n) php so snh v 0 php dch chuyn


Chn trc tip v Chn trc tip u c chi ph cho trng hp xu nht l O(n2)

Do , khng thch hp cho vic sp xp cc mng ln

D ci t, d kim li Chn trc tip tt hn Chn trc tip,

nht l khi mng c th t sn Cn c nhng thut ton hiu qu hn cho

vic sp xp cc mng ln

Nhn xt chung(Selection & Insertion sort)Nhn xt chung(Selection & Insertion sort)


Thut ton Shell sort(Shell sort Algorithm)Thut ton Shell sort(Shell sort Algorithm)c xut vo nm 1959 bi Donald L.

Shell trn tp ch Communication of the ACM

Thut ton ny ci tin hiu qu ca thutton Chn trc tip

Ph v ro cn chi ph O(n2) ca nhngthut ton sp xp trc


Shell sort AlgorithmShell sort Algorithm

tng:Chia dy ban u thnh h dy con

a0, a0+h, a0+2h,

a1, a1+h, a1+2h,

a2, a2+h, a2+2h,

Sp xp tng dy con bng cch s dng phng php Chn trc tip



Index 0 1 2 3 4 5 6 7 8 9 10 11 12Ban u 81 94 11 96 12 35 17 95 28 58 41 75 15

Chia dy thnh h=5 dy con

Index 0 1 2 3 4 5 6 7 8 9 10 11 12h=5 81 94 11 96 12 35 17 95 28 58 41 75 15

Sp xp 5 dy con bng phng php Chn trc tip

Index 0 1 2 3 4 5 6 7 8 9 10 11 12h=5 35 17 11 28 12 41 75 15 96 58 81 94 95





Index 0 1 2 3 4 5 6 7 8 9 10 11 12h=3 28 12 11 35 15 41 58 17 94 75 81 96 95

Index 0 1 2 3 4 5 6 7 8 9 10 11 12h=3 35 17 11 28 12 41 75 15 96 58 81 94 95




Index 0 1 2 3 4 5 6 7 8 9 10 11 12h=1 28 12 11 35 15 41 58 17 94 75 81 96 95


Index 0 1 2 3 4 5 6 7 8 9 10 11 12h=1 11 12 15 17 28 35 41 58 75 81 94 95 96

Kt thc !



Thut ton s dng 1 dy hk:h1, h2, h3, , ht

(*) Tnh cht dy hk: hi > hi+1 (dy gim dn) ht = 1

Dy hk gi l dy gia s (Increment sequence), dng to lp cc dy con trong mng ban u

Trong v d: h1 = 5, h2 = 3, h3 = 1



Vn : La chn dy gia s hk nh th no ?Mi dy hk tho mn tnh cht (*) u chp nhn c;

Tuy nhin, cho n nay, ngi ta ch c th ch ra rng dy hk ny tt hn dy hk kia, ch khng th xc nh c dy no l tt nht

Chi ph ca thut ton Shell sort ph thc vo 2 vn chnh l:Cch thc xy dng dy hkD liu nhp



Cc chin lc xy dng dy hk c kho st: D.Shell (1959):

h1 = [n/2], hi+1 = [hi/2], ht = 1 T.H.Hibbard (1963):

1, 3, 7, 15, . , 2k-1 (k *)N* = N \ {0} = { 1, 2, 3, 4, .}

Knuth:h1 = 1, hi = hi-1* 3 +1, v dng tai i = log2n- 11, 4, 13, 40, 121, ....

Pratt (1979):1, 2, 3, 4, 6, 8, 9, 12, 16, ..., 2p3q, (vi p, q N)


Shell sort Algorithm(Minh ha chng trnh)Shell sort Algorithm(Minh ha chng trnh)void ShellSort(int h[], int a[], int t, int n) {

for (int k=0; k


nh gi thut ton(Shell sort Algorithm)nh gi thut ton(Shell sort Algorithm) Vic phn tch gii thut ny t ra nhng vn

ton hc ht sc phc tp m trong c 1 s vn n nay vn cha c gii quyt

Ngi ta vn cha bit chn dy hk nh th no l ph hp cho ra kt qu tt nht

Mt s kt qu chng minh: Shell sort vi dy hk ca Donald Shell c s php gn

trong trng hp xu nht l O(n2) S dng dy hk ca Hibbard cn dng O(n3/2) php gn Chi ph khi dng dy hk ca Pratt l O(n(log2n)2)


Thut ton Sp xp cy(Heap sort Algorithm)Thut ton Sp xp cy(Heap sort Algorithm) c xut vo nm 1964 bi J.W.J. Williams

trn tp ch Communication of the ACM y l thut ton sp xp chm nht trong s cc

thut ton c phc tp O(n*log2n) Nhng n li t c u im v tnh n gin

ca ci t khng i hi vng qui phc tp nh ca Quicksort v khng s dng mng ph nh Mergesort


Heap sort AlgorithmNi dungHeap sort AlgorithmNi dung

nh ngha HeapBiu din Heap bng mng (array)Thao tc c bn trn HeapThut ton Heap sortnh gi thut ton


Heap sort Algorithmnh ngha HeapHeap sort Algorithmnh ngha Heap

Heap l mt cy nh phn y

Mi nt trong Heapcha mt gi tr

c th so snh vigi tr ca nt khc

19

4222127

23

45

35



c im ca Heapl gi tr ca

mi nt >= gi tr cacc nt con ca n

19

4222127

23

45

35Heap l mt

cy nh phn y



Heap l mt cy nh phn tha cc tnh cht sau sau:

L mt cy y ;Gi tr ca mi nt khng bao gi b hn gi tr

ca cc nt conH qu:Nt ln nht l ?


Heap sort AlgorithmBiu din Heap bng mngHeap sort AlgorithmBiu din Heap bng mng

Ta s lu gi tr ca cc nt trong mt array

2127

23

42

35



Gi tr ca nt gc s v tr u tin ca array

2127

23

42

35

42



Gi tr ca hai nt con ca gc c in vo hai v tr tip theo

2127

23

42

35

42 35 23



Gi tr ca hai nt hng k tip s tun t c lu li ti hai v tr tip sau

Th t lu tr trn mng c thc hin t tri sang phi

2127

23

42

35

42 35 23 27 21



Lin kt gia cc nt c hiu ngm, khng trc tip dng con tr.

Array c xem l cy ch do cch ta x l d liu trn 2127

23

42

35

42 35 23 27 21



Nu ta bit c ch s ca 1 phn t trn mng, ta s d dng xc nh c ch s ca nt cha v (cc) nt con ca n.

[0] [1] [2] [3] [4]

2127

23

42

35

42 35 23 27 21



Nt gc ch s [0]Nt cha ca nt [i] c ch s l [(i-1)/2]Cc nt con ca nt [i] (nu c) c ch s

[2i+1] v [2i+2]V d:Nt con ca nt [0] l nt [1] v nt [2]Nt cha ca nt [7] l nt [3]Nt cha ca nt [8] l nt [3]


Heap sort AlgorithmThao tc c bn trn HeapHeap sort AlgorithmThao tc c bn trn HeapHeapify - iu chnh 1 phn tNt ang xt c gi tr l

27, b hn gi tr ca nt con ca n

Tin hnh i ch vi nt con c gi tr ln nht

34

4222135

23

42

27

28


Heap sort AlgorithmThao tc c bn trn HeapHeap sort AlgorithmThao tc c bn trn Heap

Nt ang xt c gi tr l 27, b hn gi tr ca nt con ca n

Tin hnh i ch vi nt con c gi tr ln nht

34

4222127

23

42

35

28

Heapify - iu chnh 1 phn t


Heap sort AlgorithmThao tc c bn trn HeapHeap sort AlgorithmThao tc c bn trn Heap

4222134

23

42

35

2827

Heapify - iu chnh 1 phn tHon tt !


Heap sort AlgorithmThao tc c bn trn HeapHeap sort AlgorithmThao tc c bn trn Heapvoid Heapify(int a[], int n, int i) // iu chnh phn t a[i]{

int saved = a[i]; // lu li gi tr ca nt iwhile (i < n/2) { // a[i] khng phi l nt l

int child = 2*i + 1; // nt con bn tri ca a[i]if (child < n-1)

if (a[child] < a[child+1]) child ++;if (saved >= a[child]) break;a[i] = a[child];i = child;

}a[i] = saved; // Gn gi tr ca nt i vo v tr mi

}


Heap sort AlgorithmThut ton Heap sortHeap sort AlgorithmThut ton Heap sortXy dng Heap: S dng thao tc Heapify chuyn i mt mng bnh thng thnh Heap

Sp xp:Hon v phn t cui cng ca Heap vi phn

t u tin ca Heap (c gi tr ln nht)Loi b phn t cui cngThc hin thao tc Heapify iu chnh phn

t u tin


Heap sort AlgorithmThut ton Heap sort - Xy dng HeapHeap sort AlgorithmThut ton Heap sort - Xy dng Heap

Tt c cc phn t trn mng c ch s [n/2] n [n-1] u l nt l

Mi nt l c xem l Heap c mt phn tThc hin thao tc Heapify trn cc phn t

c ch s t [n/2]-1 n [0]



Thc hin Heapify vi tt c cc nt khng phi l l

Theo th t t tri sang phi

1714

23

20

12

20 12 23 14 17

Cc nt l



// Xy dng mng bnh thng a tr thnh// mt Heapvoid BuildHeap(int a[], int n){

for (int i = n/2 - 1; i >= 0; i--)Heapify(a, n, i);

}


Heap sort AlgorithmThut ton Heap sort - Sp xpHeap sort AlgorithmThut ton Heap sort - Sp xp

void HeapSort(int a[], int n){

BuildHeap(a, n);for (int i=n-1; i>=0; i--) {

Hon v a[0] vi a[i]Heapify(a, i, 0);

}}


nh gi thut ton(Heap sort Algorithm)nh gi thut ton(Heap sort Algorithm)Heap sort lun c phc tp l O(n*

log2n)Quick sort thng c phc tp l O(n*

log2n) nhng trng hp xu nht li c phc tp O(n2)

Nhn chung Quick sort nhanh hn Heap sort 2 ln nhng Heap sort li c n nh cao trong mi trng hp


Thut ton Sp xp trn(Merge sort Algorithm)Thut ton Sp xp trn(Merge sort Algorithm)L mt phhg php sp xp dng Chia tr (Divide and Conquer)

Nguyn tc Chia tr:Nu vn nh th x l ngayNu vn ln: chia thnh 2 vn nh, mi

phn bng Gii quyt tng vn nhKt hp kt qu ca nhng vn nh li vi

nhau


Merge sort AlgorithmMerge sort Algorithm

tng:Chia dy cn sp thnh 2 phn, v tr giaNu s phn t ca mi phn > 1 th

Sp sp mi phn bng Merge sortTrn 2 phn c sp li vi nhau

Thut ton Merge sort c th ci t bng qui



V d:

2 3 6 7 10 12 16 18

16 12 7 6 3 2 18 10

?



16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

V d:

Chia



16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

Chia

Chia

V d:



16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

Chia

Chia

Chia

V d:



16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

12 16 6 7 2 3 10 18

Chia

Chia

Chia

Trn

V d:



16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

12 16 6 7 2 3 10 18

6 7 12 16 2 3 10 18

Chia

Chia

Chia

Trn

Trn

V d:



16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

Chia 16 12 7 6 3 2 18 10

16 12 7 6 3 2 18 10

Trn 12 16 6 7 2 3 10 18

6 7 12 16 2 3 10 18

2 3 6 7 10 12 16 18

Chia

Chia

Trn

Trn

V d:


Merge sort Algorithm(Minh ha chng trnh)Merge sort Algorithm(Minh ha chng trnh)

void MergeSort(int a[], int Left, int Right){

int Mid; // V tr ca phn t giaif (Left < Right) { // Dy c > 1 phn t

Mid = (Left + Right)/2; // Chia thnh 2 dy MergeSort(a, Left, Mid); // Sort dy bn triMergeSort(a, Mid+1, Right); // Sort dy bn phi// Trn 2 dy li vi nhauMerge(a, Left, Mid, Right);

}} // end of MergeSort


Merge sort AlgorithmThao tc trnMerge sort AlgorithmThao tc trn

6 7 12 16 2 3 10 18

? ? ? ? ? ? ? ?

Hai dy con sp

Bng tm

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

c1 c2

d



6 7 12 16 2 3 10 18

2 ? ? ? ? ? ? ?

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

c1 c2

d

Hai dy con sp

Bng tm



6 7 12 16 2 3 10 18

2 3 ? ? ? ? ? ?

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

c1 c2

d

Hai dy con sp

Bng tm



6 7 12 16 2 3 10 18

2 3 6 ? ? ? ? ?

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

c1 c2

d

Hai dy con sp

Bng tm



6 7 12 16 2 3 10 18

2 3 6 7 ? ? ? ?

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

c1 c2

d

Hai dy con sp

Bng tm

tip tc



6 7 12 16 2 3 10 18

2 3 6 7 10 12 16 18

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

c1 c2

d

Hai dy con sp

Bng tm

Hon tt !



2 3 6 7 10 12 16 18

[0] [1] [2] [3] [4] [5] [6] [7]

[0] [1] [2] [3] [4] [5] [6] [7]

2 3 6 7 10 12 16 18Hai dy con sp

Bng tm

Copy tr li vo mng


Merge sort AlgorithmThao tc trn Minh ha chng trnhMerge sort AlgorithmThao tc trn Minh ha chng trnh

void Merge(int a[], int Left, int Mid, int Right){

// c1, c2: v tr hin ti trn dy con tri, dy con phi// d: v tr hin ti trn dy tmfor(int d=Left, int c1=Left, c2=Mid+1; (c1


Merge sort AlgorithmThao tc trn Minh ha chng trnhMerge sort AlgorithmThao tc trn Minh ha chng trnh

// Khi 1 trong 2 dy ht phn t// nu dy bn tri cn d chp vo mng tmfor( ; c1


nh gi thut ton(Merge sort Algorithm)nh gi thut ton(Merge sort Algorithm)

Trn 2 dy c kch thc n/2:O(n)

O(n).

.

.

O(n)

O(log2n)ln

Trn 4 dy c kch thc n/4:

Trn n dy c kch thc 1:


nh gi thut ton(Merge sort Algorithm)nh gi thut ton(Merge sort Algorithm)Chi ph O(n*log2n) sp xp bt k 1 dy

no S dng 1 vng nh trung gian O(n) phn tC n nh cao (khng b nh hng bi

th t ban u ca d liu)


Thut ton Sp xp nhanh(Quick sort Algorithm)Thut ton Sp xp nhanh(Quick sort Algorithm)Quick sort cng l mt thut ton chia

tr tng:Chia dy cn sp thnh 2 phnCch chia ca Quick sort khc vi cch chia

ca Merge sort: dy bn tri cha cc gi tr nh hn dy bn phi.

Thc hin vic sp xp trn tng dy con ( qui)


Quick sort AlgorithmMinh ha chng trnhQuick sort AlgorithmMinh ha chng trnh

void QuickSort(int a[], int Left, int Right){

// Ch x l khi dy c > 1 phn tif (Left < Right) {

int m1, m2;// chia dy (Left, Right) thnh 2 dy conPartition(a, Left, Right, m1, m2);QuickSort(a, Left, m1); // dy con bn triQuickSort(a, m2, Right); // dy con bn phi

}}


Quick sort AlgorithmCch chia thnh 2 dy con (Partition)Quick sort AlgorithmCch chia thnh 2 dy con (Partition)

8 32 11 754 10124 5

5

Phn t chun

Chn 1 phn t lm chun, thng ta chn phn t gia ca dy



low high

8 32 11 754 10124 5

5

Phn t chun

Bt u so snh cc phn t t 2 u ca dy vi phn t chun



low high

8 2 11 754 10124 5

tm ra c 1 phn t sai v tr pha bn phi

5

Phn t chun

33



low high

8 32 11 7512 10124 5

5

Phn t chun

tm ra c thm 1 phn t sai v tr pha bn tri



low high

8 2 11 75104 5

hon v 2 phn t cho nhau, v li tip tc

5

Phn t chun

123 312



low high

8 1211 75434 5

tm thy 2 phn t sai v tr mi

5

Phn t chun

1010 22


210


low high

8 1210 11 754 234 5

hon v cho nhau, v tip tc

5

Phn t chun



low(m2)

high(m1)

1210 11 74 234

low > high kt thc qu trnh phn chia

dy bn tri dy bn phi

8 585


Quick sort AlgorithmPartition Minh ha chng trnhQuick sort AlgorithmPartition Minh ha chng trnh

void Partition(int a[], int Left, int Right, int &m1, int &m2){

int Pivot = a[(Left+Right)/2]; // phn t chunint low = Left, high = Right;while (low < high) {

while (a[low] < Pivot) low++;while (a[high] > Pivot) high--;if (low

Quick sort AlgorithmCch chn phn t Pivot Quick sort AlgorithmCch chn phn t Pivot



nh gi thut ton(Quick sort Algorithm)nh gi thut ton(Quick sort Algorithm)Chi ph trung bnh O(n*log2n)Chi ph cho trng hp xu nht O(n2)Chi ph ty thuc vo cch chn phn t

chun:Nu chn c phn t c gi tr trung bnh, ta

s chia thnh 2 dy bng nhau; nu chn nhm phn t nh nht (hay ln nht) O(n2)

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[Slide] Chuong 1 - Cac thuat toan sap xep.pdf