Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2010 · October 2, 2010 8:58:45 AM

Quicksort Partitioning

a[i] i j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 16 K R A T E L E P U I M Q C X O S

1 12 K R A T E L E P U I M Q C X O S

1 12 K C A T E L E P U I M Q R X O S

3 9 K C A T E L E P U I M Q R X O S

3 9 K C A I E L E P U T M Q R X O S

5 6 K C A I E L E P U T M Q R X O S

5 6 K C A I E E L P U T M Q R X O S

6 5 K C A I E E L P U T M Q R X O S

6 5 E C A I E K L P U T M Q R X O S

6 5 E C A I E K L P U T M Q R X O S

Partitioning trace (array contents before and after each exchange)

initial values

scan left, scan right

exchange

scan left, scan right

exchange

scan left, scan right

exchange

scan left, scan right

!nal exchange

result

v

2

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K R A T E L E P U I M Q C X O S

i j

stop i scan because a[i] >= a[lo]

3

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K R A T E L E P U I M Q C X O S

i j

4

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K R A T E L E P U I M Q C X O S

i j

5

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K R A T E L E P U I M Q C X O S

i j

stop j scan and exchange a[i] with a[j]

6

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A T E L E P U I M Q R X O S

i j

7

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A T E L E P U I M Q R X O S

i j

8

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A T E L E P U I M Q R X O S

i j

stop i scan because a[i] >= a[lo]

9

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A T E L E P U I M Q R X O S

i j

10

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A T E L E P U I M Q R X O S

i j

11

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A T E L E P U I M Q R X O S

i j

stop j scan and exchange a[i] with a[j]

12

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E L E P U T M Q R X O S

i j

13

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E L E P U T M Q R X O S

i j

14

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E L E P U T M Q R X O S

i j

stop i scan because a[i] >= a[lo]

15

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E L E P U T M Q R X O S

i j

16

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E L E P U T M Q R X O S

i j

17

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E L E P U T M Q R X O S

i j

stop j scan and exchange a[i] with a[j]

18

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E E L P U T M Q R X O S

i j

19

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E E L P U T M Q R X O S

i

stop i scan because a[i] >= a[lo]

j

20

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

lo

K C A I E E L P U T M Q R X O S

ij

stop j scan because a[j]

21

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

When pointers cross.

• Exchange a[lo] with a[j].

lo

K C A I E E L P U T M Q R X O S

ij

pointers cross: exchange a[lo] with a[j]

22

Quicksort partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].

When pointers cross.

• Exchange a[lo] with a[j].

lo

E C A I E K L P U T M Q R X O S

hij

partitioned!

Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2010 · October 2, 2010 8:58:45 AM

Dijkstra 3-Way Partitioning

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

24

Dijkstra 3-way partitioning

lo

P A B X W P P V P D P C Y Z

hi

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

25

Dijkstra 3-way partitioning

P A B X W P P V P D P C Y Z

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

26

Dijkstra 3-way partitioning

A P B X W P P V P D P C Y Z

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

27

Dijkstra 3-way partitioning

A B P X W P P V P D P C Y Z

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

28

Dijkstra 3-way partitioning

A B P Z W P P V P D P C Y X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

29

Dijkstra 3-way partitioning

A B P Y W P P V P D P C Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

30

Dijkstra 3-way partitioning

A B P C W P P V P D P Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

31

Dijkstra 3-way partitioning

A B C P W P P V P D P Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

32

Dijkstra 3-way partitioning

A B C P P P P V P D W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

33

Dijkstra 3-way partitioning

A B C P P P P V P D W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

34

Dijkstra 3-way partitioning

A B C P P P P V P D W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

35

Dijkstra 3-way partitioning

A B C P P P P V P D W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

36

Dijkstra 3-way partitioning

A B C P P P P D P V W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

37

Dijkstra 3-way partitioning

A B C D P P P P P V W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning element a[lo].• Scan i from left to right.- (a[i] < v): exchange a[lt] with a[i] and increment both lt and i- (a[i] > v): exchange a[gt] with a[i] and decrement gt- (a[i] == v): increment i

38

Dijkstra 3-way partitioning

lo

A B C D P P P P P V W Y Z X

hi

lt gt

lt

v

gti

v

>v

Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2010 · October 2, 2010 8:58:45 AM

Bentley-McIlroy 3-Way Partitioning

40

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B X W P P V P D P C Y Z

i j

p q

41

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B X W P P V P D P C Y Z

i j

p q

42

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B X W P P V P D P C Y Z

i j

p q

43

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B X W P P V P D P C Y Z

i j

p q

hi

44

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B X W P P V P D P C Y Z

i j

p q

hi

exchange a[i] with a[j]

45

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B C W P P V P D P X Y Z

i j

p q

hi

46

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B C W P P V P D P X Y Z

i j

p q

hi

47

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B C W P P V P D P X Y Z

i j

p q

hi

exchange a[i] with a[j]

48

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P A B C P P P V P D W X Y Z

i j

p q

hi

exchange a[i] with a[p] and increment p

49

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A P P V P D W X Y Z

i j

p q

hi

50

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A P P V P D W X Y Z

i j

p q

hi

51

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A P P V P D W X Y Z

i j

p q

hi

exchange a[i] with a[j]

52

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A D P V P P W X Y Z

i j

p q

hi

exchange a[j] with a[q] and decrement q

53

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A D P V P Z W X Y P

i j

p q

hi

54

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A D P V P Z W X Y P

i j

p q

hi

55

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A D P V P Z W X Y P

i j

p q

hi

exchange a[i] with a[j]

56

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P B C A D P V P Z W X Y P

i j

p q

hi

exchange a[i] with a[p] and increment p

57

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P P C A D B V P Z W X Y P

i j

p q

hi

exchange a[j] with a[q] and decrement q

58

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P P C A D B V Y Z W X P P

i j

p q

hi

59

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P P C A D B V Y Z W X P P

i j

p q

hi

60

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P P C A D B V Y Z W X P P

i j

p q

hi

61

Bentley-McIlroy 3-way partitioning

Repeat until i and j pointers cross.

• Scan i from left to right so long as a[i] < a[lo].• Scan j from right to left so long as a[j] > a[lo].• Exchange a[i] with a[j].• If a[i] == a[lo], exchange a[i] with a[p] and increment p.• If a[j] == a[lo], exchange a[j] with a[q] and decrement q.

lo

P P P C A D B V Y Z W X P P

p q

hiij

pointers cross

62

Bentley-McIlroy 3-way partitioning

Afterwards, swap equal keys to the center.

• Scan j and p from right to left and exchange a[j] with a[p].• Scan i and q from left to right and exchange a[i] with a[q].

lo

P P P C A D B V Y Z W X P P

ij

p q

hi

exchange a[j] with a[p]

63

Bentley-McIlroy 3-way partitioning

Afterwards, swap equal keys to the center.

• Scan j and p from right to left and exchange a[j] with a[p].• Scan i and q from left to right and exchange a[i] with a[q].

lo

P P B C A D P V Y Z W X P P

ij

p q

hi

exchange a[j] with a[p]

64

Bentley-McIlroy 3-way partitioning

Afterwards, swap equal keys to the center.

• Scan j and p from right to left and exchange a[j] with a[p].• Scan i and q from left to right and exchange a[i] with a[q].

lo

P D B C A P P V Y Z W X P P

ij

p q

hi

exchange a[j] with a[p]

65

Bentley-McIlroy 3-way partitioning

Afterwards, swap equal keys to the center.

• Scan j and p from right to left and exchange a[j] with a[p].• Scan i and q from left to right and exchange a[i] with a[q].

lo

A D B C P P P V Y Z W X P P

ij

q

hi

exchange a[i] with a[q]

66

Bentley-McIlroy 3-way partitioning

Afterwards, swap equal keys to the center.

• Scan j and p from right to left and exchange a[j] with a[p].• Scan i and q from left to right and exchange a[i] with a[q].

lo

A D B C P P P P Y Z W X V P

ij

q

hi

exchange a[i] with a[q]

67

Bentley-McIlroy 3-way partitioning

Afterwards, swap equal keys to the center.

• Scan j and p from right to left and exchange a[j] with a[p].• Scan i and q from left to right and exchange a[i] with a[q].

lo

A D B C P P P P P Z W X V Y

ij hi

3-way partitioned