7.Quicksort

Hsu, Lih-Hsing

Chapter 7 P.2

Computer Theory Lab.

7.1 Description of quicksort Divide

Conquer

Combine

Chapter 7 P.3

Computer Theory Lab.

Q U I C K S O R T ( A , p , r )

1 i f p r

2 t h e n q P A R T I T I O N A p r ( , , )

3 Q U I C K S O R T ( A , p , q )

4 Q U I C K S O R T ( A , q + 1 , r )

Chapter 7 P.4

Computer Theory Lab.

Partition(A, p, r)

1 x A[r]

2 i p – 1

3 for j p to r -1

4 do if A[j] ≤ x

5 then i i + 1

6 exchange A[i] A[j]

7 exchange A[i +1] A[r]

8 return i +1

Chapter 7 P.5

Computer Theory Lab.

At the beginning of each iteration of the loop of lines 3-6, for any array index k,

1. if p ≤ k ≤ i, then A[k] ≤ x.

2. if i + 1 ≤ k≤ j -1, then A[k] > x.

3. if k = r, then A[k] = x.

Chapter 7 P.6

Computer Theory Lab.

The operation of Partition on a sample array

Chapter 7 P.7

Computer Theory Lab.

Two cases for one iteration of procedure Partition

Complexity: Partition on A[p…r] is (n) where n = r –p +1

Chapter 7 P.8

Computer Theory Lab.

7.2 Performance of quicksort W o r s t - c a s e p a r t i t i o n :

T n T n n

k k nk

n

k

n

( ) ( ) ( )

( ) ( ) ( )

1

1 1

2

B e s t - c a s e p a r t i t i o n :

T n T n n

T n n n

( ) ( / ) ( )

( ) ( l o g )

2 2

Chapter 7 P.9

Computer Theory Lab.

Balanced partition T n n n( ) ( log )

)log()(

)()10/()10/9()(

nnnT

nnTnTnT

Chapter 7 P.10

Computer Theory Lab.

Intuition for the average case T(n) = (n logn)

Chapter 7 P.11

Computer Theory Lab.

7.3 Randomized versions of partition

R A N D O M I Z E D _ P A R T I T I O N ( A , p , r )

1 i R A N D O M p r ( , )

2 e x c h a n g e A p A i[ ] [ ]

3 r e t u r n P A R T I T I O N ( A , p , r )

R A N D O M I Z E D _ Q U I C K S O R T ( A , p , r )

1 i f p r

2 t h e n

q R A N D O M I Z E D P A R T I T I O N A p r _ ( , , )

3 R A N D O M I Z E D _ Q U I C K S O R T ( A , p , q )

4 R A N D O M I Z E D _ Q U I C K S O R T ( A , q + 1 , r )

Chapter 7 P.12

Computer Theory Lab.

7.4 Analysis of quicksort7.4.1 Worst-case analysis

)())1()(()( max10

nqnTqTnTnq

guess T n cn( ) 2

2

2

22

10

22

10

)()1(2

)())1((

)())1(()(

max

max

cn

nnccn

nqnqc

nqnccqnT

nq

nq

pick the constant c large enough so that the 2 1c n( ) term dominates the

( )n term.

T n n( ) ( ) 2

Chapter 7 P.13

Computer Theory Lab.

7.4-2 Show that 22 )( qnq achieves a maximum over

1,.....,2,1 nq when 1q or 1nq

ans: 先令 22 )()( qnqqf

一次微分: nqqnqqf 24)(22)('

令 0)(' qf 024 nq 2

nq(極小值)

二次微分: 4)('' qf (開口向上)

因為 11 nq 所以 2)1(1)1()1( nnff (相對極大值)

nn/20

Chapter 7 P.14

Computer Theory Lab.

7.4.2 Expected running time Running time and comparsions Lemma 7.1

Let X be the number of comparisons performed in line 4 of partition over the entire execution of Quicksort on an n-element array. Then the running rime of Quicksort is O(n+X)

Chapter 7 P.15

Computer Theory Lab.

we define

{zi is compared to zj},

{zi is compared to zj}

IX ij

.1

1 1

n

i

n

ijijXX

1

1 1

1

1 1

1

1 1

Pr

][

n

i

n

ij

n

i

n

ijij

n

i

n

ijij

XE

XEXE

Chapter 7 P.16

Computer Theory Lab.

Pr{zi is compared to zj} = Pr{zi or zj is first pivot chosen from Zij}

= Pr{zi is first pivot chosen from Zij}

+ Pr{zj is first pivot chosen from Zij}

1

1

1

1

ijij

1

2

ij

1

1 1

.1

2][

n

i

n

ij ijXE

Chapter 7 P.17

Computer Theory Lab.

)lg(

)(lg

2

1

2

1

2][

1

1

1

1 1

1

1 1

1

1 1

nnO

nO

k

k

ijXE

n

i

n

i

n

k

n

i

in

k

n

i

n

ij

Chapter 7 P.18

Computer Theory Lab.

another analysis

T n

nT T n T q T n q n

q

n

( )

( ( ) ( ) ( ( ) ( ) ) ( )

11 1

1

1

T

T n O n nT T n O n

( )

( ) ( )( ( ) ( ) ) ( )

1 1

1

11 1

2

T n

nT T n T q T n q n

nT q T n q n

nT k n

q

n

q

n

k

n

( )

( ( ) ( ) ( ( ) ( ) ) ( )

( ( ) ( ) ) ( )

( ( ) ) ( )

11 1

1

2

1

1

1

1

1

1

Chapter 7 P.19

Computer Theory Lab.

g u e s s bnannT log)(

T nn

a k k b n

an

k kb

nn n

k

n

k

n

( ) ( l o g ) ( )

l o g ( ) ( )

2

2 21

1

1

1

1

W e w i l l p r o v e k kn n n

k

nl o g

l o g

2

1

1 2

2 8

Chapter 7 P.20

Computer Theory Lab.

bnan

anbnbnan

nban

nan

nn

nbnnn

na

nT

log

)4

)((log

)(24

log

)()1(2

)8

log21

(2

)(2

2

C h o o s e a l a r g e e n o u g h s o t h a t a n

n b4

( ) .

T n O n n( ) ( l o g ) .

Chapter 7 P.21

Computer Theory Lab.

.2 if82

log2

)12

(2

1

2

log)1(

log

log)1(log

logloglog

22

12/

1

1

1

1

2/

12/

1

1

2/

1

1

12/

1

n

nnn

nnnnn

kkn

knkn

kkkkkk

n

k

n

k

n

nk

n

k

n

nk

n

k

n

k

A n o t h e r a p p r o a c h : U s i n g

x x d x x x xl n l n 12

14

2 2