[PPT] - Recursion [Andersen, Gries, Lee, Marschner, Van Loan, White] PowerPoint Presentation

SLIDE 1

Recursion

Lecture 15

CS 1110:

Introduction to Computing Using Python

[Andersen, Gries, Lee, Marschner, Van Loan, White]

SLIDE 2

Announcements: Prelim 1

Graded and released
Mean: 81 out of 104 (78%)
Can pick up your exam in homework handback room
Need Cornell ID
Suggest printing your netid on paper
Do not discuss exam with people taking makeups.
Regrade requests: we will send email to you

10/13/16 Recursion 2

SLIDE 3

Announcements: Assignment 3

Released.
Due: Thursday, March 30th, 11:59pm
Recommendation: follow milestone deadlines.
You MUST acknowledge help from others
We run software analyzers to detect similar programs
Have had some academic integrity violations so far
Not a recursion assignment!

10/13/16 Recursion 3

SLIDE 4

Announcement: Lab 8

Out.
Not a recursion lab!

10/13/16 Recursion 4

SLIDE 5

Recursion

Recursive Definition:

A definition that is defined in terms of itself

10/13/16 Recursion 5

SLIDE 6

A Mathematical Example: Factorial

Non-recursive definition:

n! = n × n-1 × … × 2 × 1 = n (n-1 × … × 2 × 1)

Recursive definition:

n! = n (n-1)! 0! = 1

10/13/16 Recursion 6

for n ≥ 0 Recursive case Base case What happens if there is no base case?

SLIDE 7

Recursion

Recursive Definition:

A definition that is defined in terms of itself

Recursive Function:

A function that calls itself (directly or indirectly)

10/13/16 Recursion 7

SLIDE 8

Recursive Call Frames

10/13/16 Recursion 8

factorial 1

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

1 2 3

SLIDE 9

Recursive Call Frames

10/13/16 Recursion 9

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

1 2 3

SLIDE 10

Recursion

10/13/16 Recursion 10

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

1 2 3

Now what? Each call is a new frame.

SLIDE 11

A:

What happens next?

10/13/16 Recursion 11

factorial 1, 3

n

3 factorial 1

n

2 factorial 1, 3, 1

n

3 2

 B:

factorial 1

n

3

C:

factorial 1, 3, 1

n

3 2

 D:

factorial 1

n

2 ERASE FRAME factorial 1, 3

n

3

CORRECT

SLIDE 12

Recursive Call Frames

10/13/16 Recursion 12

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1

SLIDE 13

Recursive Call Frames

10/13/16 Recursion 13

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3

SLIDE 14

Recursive Call Frames

10/13/16 Recursion 14

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 1

SLIDE 15

Recursive Call Frames

10/13/16 Recursion 15

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 1, 3

SLIDE 16

Recursive Call Frames

10/13/16 Recursion 16

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 1, 3 factorial

n

1

SLIDE 17

Recursive Call Frames

10/13/16 Recursion 17

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 1, 3 factorial

n

1, 2

SLIDE 18

Recursive Call Frames

10/13/16 Recursion 18

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 1, 3 factorial

n

RETURN 1 1, 2

SLIDE 19

Recursive Call Frames

10/13/16 Recursion 19

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 1, 3 factorial

n

RETURN 1 1, 2

SLIDE 20

Recursive Call Frames

10/13/16 Recursion 20

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 RETURN 1 1, 3 factorial

n

RETURN 1 1, 2

SLIDE 21

Recursive Call Frames

10/13/16 Recursion 21

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2

1 2 3

1, 3 factorial

n

1 RETURN 1 1, 3 factorial

n

RETURN 1 1, 2

SLIDE 22

Recursive Call Frames

10/13/16 Recursion 22

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2 RETURN

1 2 3

2 1, 3 factorial

n

1 RETURN 1 1, 3 factorial

n

RETURN 1 1, 2

SLIDE 23

Recursive Call Frames

10/13/16 Recursion 23

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

factorial

n

2 RETURN

1 2 3

2 1, 3 factorial

n

1 RETURN 1 1, 3 factorial

n

RETURN 1 1, 2

SLIDE 24

Recursive Call Frames

10/13/16 Recursion 24

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

RETURN factorial

n

2 RETURN

1 2 3

2 1, 3 factorial

n

1 RETURN 1 1, 3 factorial

n

RETURN 1 1, 2 6

SLIDE 25

Recursive Call Frames

10/13/16 Recursion 25

factorial 1, 3

n

3

def factorial(n): """Returns: factorial of n. Pre: n ≥ 0 an int""" if n == 0: return 1 return n*factorial(n-1) Call: factorial(3)

RETURN factorial

n

2 RETURN

1 2 3

2 1, 3 factorial

n

1 RETURN 1 1, 3 factorial

n

RETURN 1 1, 2 6

SLIDE 26

Example: Fibonnaci Sequence

Sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ...

a0 a1 a2 a3 a4 a5 a6

Get the next number by adding previous two
What is a8?

10/13/16 Recursion 26

A: a8 = 21 B: a8 = 29 C: a8 = 34 D: None of these.

SLIDE 27

Example: Fibonnaci Sequence

Sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ...

a0 a1 a2 a3 a4 a5 a6

Get the next number by adding previous two
What is a8?

10/13/16 Recursion 27

A: a8 = 21 B: a8 = 29 C: a8 = 34 D: None of these. correct

SLIDE 28

Example: Fibonnaci Sequence

Sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ...

a0 a1 a2 a3 a4 a5 a6

Get the next number by adding previous two
What is a8?
Recursive definition:
an = an-1 + an-2

Recursive Case

a0 = 1

Base Case

a1 = 1

(another) Base Case

10/13/16 Recursion 28

Why did we need two base cases this time?

SLIDE 29

Fibonacci as a Recursive Function

def fibonacci(n):

"""Returns: Fibonacci no. an Precondition: n ≥ 0 an int""" if n <= 1: return 1 return (fibonacci(n-1)+ fibonacci(n-2))

10/13/16 Recursion 29

Recursive case Base case(s) Handles both base cases in one conditional.

SLIDE 30

Fibonacci as a Recursive Function

def fibonacci(n):

"""Returns: Fibonacci no. an Precondition: n ≥ 0 an int""" if n <= 1: return 1 return (fibonacci(n-1)+ fibonacci(n-2))

10/13/16 Recursion 30

n fibonacci 3 5 n fibonacci 1 4 n fibonacci 1 3

SLIDE 31

Recursion vs Iteration

Recursion is provably equivalent to iteration
Iteration includes for-loop and while-loop (later)
Anything can do in one, can do in the other
But some things are easier with recursion
And some things are easier with iteration
Will not teach you when to choose recursion
We just want you to understand the technique

10/13/16 Recursion 31

SLIDE 32

Recursion is best for Divide and Conquer

Goal: Solve problem P on a piece of data

10/13/16 Recursion 32

data

SLIDE 33

Recursion is best for Divide and Conquer

Goal: Solve problem P on a piece of data

10/13/16 Recursion 33

data

Idea: Split data into two parts and solve problem

data 1 data 2

Solve Problem P Solve Problem P

SLIDE 34

Recursion is best for Divide and Conquer

Goal: Solve problem P on a piece of data

10/13/16 Recursion 34

data

Idea: Split data into two parts and solve problem

data 1 data 2

Solve Problem P Solve Problem P Combine Answer!

SLIDE 35

Divide and Conquer Example

Count the number of 'e's in a string:

10/13/16 Recursion 35

p e n n e

Two 'e's

p e n n e

One 'e' One 'e'

SLIDE 36

Divide and Conquer Example

Count the number of 'e's in a string:

10/13/16 Recursion 36

p e n n e

Two 'e's

p e n n e

Zero 'e's Two 'e's

SLIDE 37

Three Steps for Divide and Conquer

1. Decide what to do on “small” data
Some data cannot be broken up
Have to compute this answer directly
2. Decide how to break up your data
Both “halves” should be smaller than whole
Often no wrong way to do this (next lecture)
3. Decide how to combine your answers
Assume the smaller answers are correct
Combining them should give bigger answer

10/13/16 Recursion 37

SLIDE 38

Divide and Conquer Example

def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Combine the result return left+right

“Short-cut” for

if s[0] == 'e’: return 1 else: return 0

10/13/16 Recursion 38

p e n n e

2 +

s[0] s[1:]

SLIDE 39

Divide and Conquer Example

def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Combine the result return left+right

“Short-cut” for

if s[0] == 'e’: return 1 else: return 0

10/13/16 Recursion 39

p e n n e

2 +

s[0] s[1:]

SLIDE 40

Divide and Conquer Example

def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Combine the result return left+right

“Short-cut” for

if s[0] == 'e’: return 1 else: return 0

10/13/16 Recursion 40

p e n n e

2 +

s[0] s[1:]

SLIDE 41

Divide and Conquer Example

def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Combine the result return left+right

“Short-cut” for

if s[0] == 'e’: return 1 else: return 0

10/13/16 Recursion 41

p e n n e

2 +

s[0] s[1:]

SLIDE 42

Divide and Conquer Example

def num_es(s): """Returns: # of 'e's in s""" # 1. Handle small data if s == '': return 0 elif len(s) == 1: return 1 if s[0] == 'e' else 0 # 2. Break into two parts left = num_es(s[0]) right = num_es(s[1:]) # 3. Combine the result return left+right

10/13/16 Recursion 42

Base Case Recursive Case

SLIDE 43

Exercise: Remove Blanks from a String

def deblank(s): """Returns: s but with its blanks removed"""

1. Decide what to do on “small” data
If it is the empty string, nothing to do

if s == '': return s

If it is a single character, delete it if a blank

if s == ' ': # There is a space here return '' # Empty string else: return s

10/13/16 Recursion 43

SLIDE 44

Exercise: Remove Blanks from a String

def deblank(s): """Returns: s but with its blanks removed"""

2. Decide how to break it up

left = deblank(s[0]) # A string with no blanks right = deblank(s[1:]) # A string with no blanks

3. Decide how to combine the answer

return left+right # String concatenation

10/13/16 Recursion 44

SLIDE 45

Putting it All Together

def deblank(s): """Returns: s w/o blanks""" if s == '': return s elif len(s) == 1: return '' if s[0] == ' ' else s left = deblank(s[0]) right = deblank(s[1:]) return left+right

10/13/16 Recursion 45

Handle small data Break up the data Combine answers

SLIDE 46

Putting it All Together

def deblank(s): """Returns: s w/o blanks""" if s == '': return s elif len(s) == 1: return '' if s[0] == ' ' else s left = deblank(s[0]) right = deblank(s[1:]) return left+right

10/13/16 Recursion 46

Base Case Recursive Case

SLIDE 47

Following the Recursion

a b c deblank

10/13/16 Recursion 47

SLIDE 48

Following the Recursion

a b c deblank

10/13/16 Recursion 48

deblank a b c deblank stops (base case) a deblank stops (base case) b c deblank …

SLIDE 49

Following the Recursion

a b c deblank a b c deblank

10/13/16 Recursion 49

SLIDE 50

Following the Recursion

a b c a deblank a b c deblank b c deblank

10/13/16 Recursion 50

SLIDE 51

Following the Recursion

a b c a deblank a b c deblank b c deblank b c deblank

10/13/16 Recursion 51

SLIDE 52

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank

10/13/16 Recursion 52

SLIDE 53

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank

10/13/16 Recursion 53

SLIDE 54

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c

SLIDE 55

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c c

SLIDE 56

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c c c



SLIDE 57

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c c c c b



SLIDE 58

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c c c c b c b

 

SLIDE 59

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c c c c b c b c b a

 

SLIDE 60

Following the Recursion

a b c a b deblank a b c deblank b c deblank b c deblank c deblank c deblank c c c c b c b c b a c b a

  

SLIDE 61

Following the Recursion

a b c a b c c c c b c b c b a c b a c b a deblank a b c deblank b c deblank b c deblank c deblank c deblank

  

SLIDE 62

Tower of Hanoi

Three towers: left, middle, and right
n disks of unique sizes on left
Goal: move all disks from left to right
Cannot put a larger disk on top of a smaller disk

10/13/16 Recursion 62

4

left middle right

3 2 1

SLIDE 63

1 Disc

1. Move from left to right

10/13/16 Recursion 63

1

left middle right

SLIDE 64

1 Disc

1. Move from left to right

10/13/16 Recursion 64

1

left middle right

SLIDE 65

2 Discs

1. Move from left to middle

10/13/16 Recursion 65

left middle right

2 1

SLIDE 66

2 Discs

10/13/16 Recursion 66

left middle right

2 1

1. Move from left to middle
2. Move from left to right

SLIDE 67

2 Discs

10/13/16 Recursion 67

left middle right

2 1

1. Move from left to middle
2. Move from left to right
3. Move from middle to right

SLIDE 68

2 Discs

10/13/16 Recursion 68

left middle right

2 1

1. Move from left to middle
2. Move from left to right
3. Move from middle to right

SLIDE 69

3 Discs

1. Move from left to right

10/13/16 Recursion 70

left middle right

3 2 1

SLIDE 70

3 Discs

1. Move from left to right
2. Move from left to middle

10/13/16 Recursion 71

left middle right

3 2 1

SLIDE 71

3 Discs

1. Move from left to right
2. Move from left to middle
3. Move from right to middle

10/13/16 Recursion 72

left middle right

3 2 1

SLIDE 72

3 Discs

1. Move from left to right
2. Move from left to middle
3. Move from right to middle
4. Move from left to right

10/13/16 Recursion 73

left middle right

3 2 1

SLIDE 73

3 Discs

1. Move from left to right
2. Move from left to middle
3. Move from right to middle
4. Move from left to right
5. Move from middle to left

10/13/16 Recursion 74

left middle right

3 2 1

SLIDE 74

3 Discs

1. Move from left to right
2. Move from left to middle
3. Move from right to middle
4. Move from left to right
5. Move from middle to left
6. Move from middle to right

10/13/16 Recursion 75

left middle right

3 2 1

SLIDE 75

3 Discs

1. Move from left to right
2. Move from left to middle
3. Move from right to middle
4. Move from left to right
5. Move from middle to left
6. Move from middle to right
7. Move from left to right

10/13/16 Recursion 76

left middle right

3 2 1

SLIDE 76

3 Discs

1. Move from left to right
2. Move from left to middle
3. Move from right to middle
4. Move from left to right
5. Move from middle to left
6. Move from middle to right
7. Move from left to right

10/13/16 Recursion 77

left middle right

3 2 1

SLIDE 77

4

4 Discs: High-level Idea

10/13/16 Recursion 78

left middle right

3 2 1

SLIDE 78

4

4 Discs: High-level Idea

10/13/16 Recursion 79

left middle right

3 2 1

Plan: move top three disks

from left to middle

SLIDE 79

4

4 Discs: High-level Idea

10/13/16 Recursion 80

left middle right

3 2 1

Plan: move top three disks

from left to middle

Move: largest disk from left

to right

SLIDE 80

4

4 Discs: High-level Idea

10/13/16 Recursion 81

left middle right

3 2 1

Plan: move top three disks

from left to middle

Move: largest disk from left

to right

Plan: move top three disks

from middle to right

SLIDE 81

4

4 Discs: High-level Idea

10/13/16 Recursion 82

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

SLIDE 82

4

4 Discs: High-level Idea

10/13/16 Recursion 83

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

SLIDE 83

4

4 Discs: High-level Idea

10/13/16 Recursion 84

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right

SLIDE 84

4

4 Discs: High-level Idea

10/13/16 Recursion 85

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right
Plan: move disks 1 and 2 from

right to middle

SLIDE 85

4

4 Discs: High-level Idea

10/13/16 Recursion 86

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right
Plan: move disks 1 and 2 from

right to middle

SLIDE 86

4

4 Discs: High-level Idea

10/13/16 87

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right
Plan: move disks 1 and 2 from

right to middle

Move: disk 4 from left to

right

Recursion

SLIDE 87

4

4 Discs: High-level Idea

10/13/16 88

left middle right

3 2 1

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right
Plan: move disks 1 and 2 from

right to middle

Move: disk 4 from left to

right

Plan: move disks 1, 2, and 3

from middle to right

SLIDE 88

4 Discs: High-level Idea

10/13/16 89

left middle

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right
Plan: move disks 1 and 2 from

right to middle

Move: disk 4 from left to

right

Plan: move disks 1, 2, and 3

from middle to right

4

right

3 2 1

SLIDE 89

Observation: Plans within a Plan

10/13/16 90

Plan: move disks 1, 2, and 3

from left to middle

Plan: move disks 1 and 2 from

left to right

Move: disk 3 from left to right
Plan: move disks 1 and 2 from

right to middle

Move: disk 4 from left to

right

Plan: move disks 1, 2, and 3

from middle to right High-level plan Low-level plan

SLIDE 90

General Pattern

To move n disks from source to target:

(source, other, and target can be any permutation of left, middle and right)

1. Plan: move disks 1, …, n-1 from source to other 2. Move: disk n to from source to target 3. Plan: move disks 1, …, n-1 from other to target

10/13/16 Recursion 91

4 3 2 1

disks 1, …, n-1 disk n source

ther

target

ther disks?