Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2011 · September 24, 2011 8:19:02 AM

AlgorithmsF O U R T H E D I T I O N

R O B E R T S E D G E W I C K K E V I N W A Y N E

2.3 QUICKSORT PARTITIONING DEMO

‣ Sedgewick 2-way‣ Dijkstra 3-way‣ Bentley-McIlroy 3-way

Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2011 · September 24, 2011 8:19:02 AM

AlgorithmsF O U R T H E D I T I O N

R O B E R T S E D G E W I C K K E V I N W A Y N E

SEDGEWICK 2-WAY PARTITIONING

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

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

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

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 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]

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

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

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

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

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 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]

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

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

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

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

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 L E P U T M Q R X O S

i j

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

i

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

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].

lo

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

ij

stop j scan because 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

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]

23

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 © 2002–2011 · September 24, 2011 8:19:02 AM

AlgorithmsF O U R T H E D I T I O N

R O B E R T S E D G E W I C K K E V I N W A Y N E

DIJKSTRA 3-WAY PARTITIONING

• Let v be partitioning item 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

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 item 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

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

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 X W P P V P D P C Y Z

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 Z W P P V P D P C Y X

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 Y W P P V P D P C Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 P C W P P V P D P Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 W P P V P D P Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 item 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 item 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 item 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 V P D W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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 P P P P D P V W Y Z X

lt gt

lt

v

gti

v

>v

• Let v be partitioning item 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

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 item 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

39

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 © 2002–2011 · September 24, 2011 8:19:02 AM

AlgorithmsF O U R T H E D I T I O N

R O B E R T S E D G E W I C K K E V I N W A Y N E

BENTLEY-MCILROY 3-WAY PARTITIONING

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

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

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 X W P P V P D P C Y Z

i j

p q

hi

exchange a[i] with a[j]

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

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 W P P V P D P X Y Z

i j

p q

hi

exchange a[i] with a[j]

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 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

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

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 P P V P D W X Y Z

i j

p q

hi

exchange a[i] with a[j]

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 P W X Y Z

i j

p q

hi

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

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

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[j]

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 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

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 P Z W X Y P

i j

p q

hi

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

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

i j

p q

hi

62

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

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 P C A D B 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 P B C A D 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

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

ij

p q

hi

exchange a[j] with a[p]

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 V Y Z W X P 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 Y Z W X V P

ij

q

hi

exchange a[i] with a[q]

68

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