61A Lecture 6 Announcements Recursive Functions Recursive - - PowerPoint PPT Presentation

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61A Lecture 6 Announcements Recursive Functions Recursive - - PowerPoint PPT Presentation

Oct 31, 2022 199 likes •1.08k views

61A Lecture 6 Announcements Recursive Functions Recursive Functions 4 Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly 4 Recursive Functions Definition:

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SLIDE 1

61A Lecture 6

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SLIDE 2

Announcements

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SLIDE 3

Recursive Functions

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SLIDE 4

Recursive Functions

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SLIDE 5

Recursive Functions

Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly

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SLIDE 6

Recursive Functions

Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly Implication: Executing the body of a recursive function may require applying that function

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SLIDE 7

Recursive Functions

Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly Implication: Executing the body of a recursive function may require applying that function

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SLIDE 8

Recursive Functions

Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly Implication: Executing the body of a recursive function may require applying that function

Drawing Hands, by M. C. Escher (lithograph, 1948)

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SLIDE 9

Digit Sums

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2+0+1+5 = 8

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SLIDE 10

Digit Sums

  • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9

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2+0+1+5 = 8

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SLIDE 11

Digit Sums

  • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9
  • Useful for typo detection!

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2+0+1+5 = 8

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SLIDE 12

Digit Sums

  • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9
  • Useful for typo detection!

5

The Bank of 61A 1234 5678 9098 7658

OSKI THE BEAR

2+0+1+5 = 8

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SLIDE 13

Digit Sums

  • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9
  • Useful for typo detection!

5

The Bank of 61A 1234 5678 9098 7658

OSKI THE BEAR

A checksum digit is a function of all the other digits; It can be computed to detect typos 2+0+1+5 = 8

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SLIDE 14

Digit Sums

  • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9
  • Useful for typo detection!

5

The Bank of 61A 1234 5678 9098 7658

OSKI THE BEAR

A checksum digit is a function of all the other digits; It can be computed to detect typos

  • Credit cards actually use the Luhn algorithm, which we'll implement after digit_sum

2+0+1+5 = 8

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SLIDE 15

Sum Digits Without a While Statement

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SLIDE 16

Sum Digits Without a While Statement

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def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10

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SLIDE 17

Sum Digits Without a While Statement

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def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n."""

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SLIDE 18

Sum Digits Without a While Statement

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def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n

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SLIDE 19

Sum Digits Without a While Statement

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def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n)

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SLIDE 20

Sum Digits Without a While Statement

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def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 21

The Anatomy of a Recursive Function

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 22

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 23

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 24

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 25

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases

7

def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 26

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases
  • Base cases are evaluated without recursive calls

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 27

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases
  • Base cases are evaluated without recursive calls

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 28

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases
  • Base cases are evaluated without recursive calls
  • Recursive cases are evaluated with recursive calls

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 29

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases
  • Base cases are evaluated without recursive calls
  • Recursive cases are evaluated with recursive calls

7

def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 30

The Anatomy of a Recursive Function

  • The def statement header is similar to other functions
  • Conditional statements check for base cases
  • Base cases are evaluated without recursive calls
  • Recursive cases are evaluated with recursive calls

(Demo)

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def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

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SLIDE 31

Recursion in Environment Diagrams

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SLIDE 32

Recursion in Environment Diagrams

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Interactive Diagram

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SLIDE 33

Recursion in Environment Diagrams

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(Demo) Interactive Diagram

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SLIDE 34

Recursion in Environment Diagrams

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(Demo) Interactive Diagram

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SLIDE 35

Recursion in Environment Diagrams

  • The same function fact is called

multiple times

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(Demo) Interactive Diagram

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SLIDE 36

Recursion in Environment Diagrams

  • The same function fact is called

multiple times

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(Demo) Interactive Diagram

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SLIDE 37

Recursion in Environment Diagrams

  • The same function fact is called

multiple times

  • Different frames keep track of the

different arguments in each call

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(Demo) Interactive Diagram

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SLIDE 38

Recursion in Environment Diagrams

  • The same function fact is called

multiple times

  • Different frames keep track of the

different arguments in each call

  • What n evaluates to depends upon 


the current environment

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(Demo) Interactive Diagram

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SLIDE 39

Recursion in Environment Diagrams

  • The same function fact is called

multiple times

  • Different frames keep track of the

different arguments in each call

  • What n evaluates to depends upon 


the current environment

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(Demo) Interactive Diagram

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SLIDE 40

Recursion in Environment Diagrams

  • The same function fact is called

multiple times

  • Different frames keep track of the

different arguments in each call

  • What n evaluates to depends upon 


the current environment

  • Each call to fact solves a simpler

problem than the last: smaller n

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(Demo) Interactive Diagram

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SLIDE 41

Iteration vs Recursion

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SLIDE 42

Iteration vs Recursion

Iteration is a special case of recursion

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SLIDE 43

4! = 4 · 3 · 2 · 1 = 24

Iteration vs Recursion

Iteration is a special case of recursion

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SLIDE 44

4! = 4 · 3 · 2 · 1 = 24

Iteration vs Recursion

Iteration is a special case of recursion Using while:

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SLIDE 45

4! = 4 · 3 · 2 · 1 = 24

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total Using while:

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SLIDE 46

4! = 4 · 3 · 2 · 1 = 24

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total Using while: Using recursion:

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SLIDE 47

4! = 4 · 3 · 2 · 1 = 24

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion:

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SLIDE 48

4! = 4 · 3 · 2 · 1 = 24

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion: Math:

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SLIDE 49

4! = 4 · 3 · 2 · 1 = 24

n! =

n

Y

k=1

k

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion: Math:

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SLIDE 50

4! = 4 · 3 · 2 · 1 = 24

n! =

n

Y

k=1

k n! = ( 1 if n = 0 n · (n − 1)!

  • therwise

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion: Math:

10

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SLIDE 51

4! = 4 · 3 · 2 · 1 = 24

n! =

n

Y

k=1

k n! = ( 1 if n = 0 n · (n − 1)!

  • therwise

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion: Math: Names:

10

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SLIDE 52

4! = 4 · 3 · 2 · 1 = 24

n! =

n

Y

k=1

k n! = ( 1 if n = 0 n · (n − 1)!

  • therwise

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion:

n, total, k, fact_iter

Math: Names:

10

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SLIDE 53

4! = 4 · 3 · 2 · 1 = 24

n! =

n

Y

k=1

k n! = ( 1 if n = 0 n · (n − 1)!

  • therwise

Iteration vs Recursion

Iteration is a special case of recursion def fact_iter(n): total, k = 1, 1 while k <= n: total, k = total*k, k+1 return total def fact(n): if n == 0: return 1 else: return n * fact(n-1) Using while: Using recursion:

n, total, k, fact_iter

Math: Names:

n, fact

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SLIDE 54

Verifying Recursive Functions

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SLIDE 55

The Recursive Leap of Faith

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SLIDE 56

The Recursive Leap of Faith

Photo by Kevin Lee, Preikestolen, Norway

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SLIDE 57

The Recursive Leap of Faith

Photo by Kevin Lee, Preikestolen, Norway

def fact(n): if n == 0: return 1 else: return n * fact(n-1)

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SLIDE 58

The Recursive Leap of Faith

Is fact implemented correctly?

Photo by Kevin Lee, Preikestolen, Norway

def fact(n): if n == 0: return 1 else: return n * fact(n-1)

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SLIDE 59

The Recursive Leap of Faith

Is fact implemented correctly? 1. Verify the base case

Photo by Kevin Lee, Preikestolen, Norway

def fact(n): if n == 0: return 1 else: return n * fact(n-1)

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SLIDE 60

The Recursive Leap of Faith

Is fact implemented correctly? 1. Verify the base case 2. Treat fact as a functional abstraction!

Photo by Kevin Lee, Preikestolen, Norway

def fact(n): if n == 0: return 1 else: return n * fact(n-1)

12

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SLIDE 61

The Recursive Leap of Faith

Is fact implemented correctly? 1. Verify the base case 2. Treat fact as a functional abstraction! 3. Assume that fact(n-1) is correct

Photo by Kevin Lee, Preikestolen, Norway

def fact(n): if n == 0: return 1 else: return n * fact(n-1)

12

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SLIDE 62

The Recursive Leap of Faith

Is fact implemented correctly? 1. Verify the base case 2. Treat fact as a functional abstraction! 3. Assume that fact(n-1) is correct 4. Verify that fact(n) is correct

Photo by Kevin Lee, Preikestolen, Norway

def fact(n): if n == 0: return 1 else: return n * fact(n-1)

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SLIDE 63

Mutual Recursion

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SLIDE 64

The Luhn Algorithm

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SLIDE 65

The Luhn Algorithm

Used to verify credit card numbers

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SLIDE 66

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

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SLIDE 67

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

14

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SLIDE 68

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

  • Second: Take the sum of all the digits

14

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SLIDE 69

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

  • Second: Take the sum of all the digits

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1 3 8 7 4 3

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SLIDE 70

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

  • Second: Take the sum of all the digits

14

1 3 8 7 4 3 2 3 1+6=7 7 8 3

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SLIDE 71

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

  • Second: Take the sum of all the digits

14

1 3 8 7 4 3 2 3 1+6=7 7 8 3 = 30

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SLIDE 72

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

  • Second: Take the sum of all the digits

14

1 3 8 7 4 3 2 3 1+6=7 7 8 3 The Luhn sum of a valid credit card number is a multiple of 10 = 30

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SLIDE 73

The Luhn Algorithm

Used to verify credit card numbers From Wikipedia: http://en.wikipedia.org/wiki/Luhn_algorithm

  • First: From the rightmost digit, which is the check digit, moving left, double the value
  • f every second digit; if product of this doubling operation is greater than 9 (e.g., 7 *

2 = 14), then sum the digits of the products (e.g., 10: 1 + 0 = 1, 14: 1 + 4 = 5)

  • Second: Take the sum of all the digits

14

1 3 8 7 4 3 2 3 1+6=7 7 8 3 The Luhn sum of a valid credit card number is a multiple of 10 = 30 (Demo)

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SLIDE 74

Recursion and Iteration

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SLIDE 75

Converting Recursion to Iteration

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SLIDE 76

Converting Recursion to Iteration

Can be tricky: Iteration is a special case of recursion.

16

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SLIDE 77

Converting Recursion to Iteration

Can be tricky: Iteration is a special case of recursion. Idea: Figure out what state must be maintained by the iterative function.

16

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SLIDE 78

def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

Converting Recursion to Iteration

Can be tricky: Iteration is a special case of recursion. Idea: Figure out what state must be maintained by the iterative function.

16

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SLIDE 79

def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

Converting Recursion to Iteration

Can be tricky: Iteration is a special case of recursion. Idea: Figure out what state must be maintained by the iterative function.

16

What's left to sum

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SLIDE 80

def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

Converting Recursion to Iteration

Can be tricky: Iteration is a special case of recursion. Idea: Figure out what state must be maintained by the iterative function. A partial sum

16

What's left to sum

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SLIDE 81

def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last

Converting Recursion to Iteration

Can be tricky: Iteration is a special case of recursion. Idea: Figure out what state must be maintained by the iterative function. A partial sum

16

(Demo) What's left to sum

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SLIDE 82

Converting Iteration to Recursion

17

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SLIDE 83

Converting Iteration to Recursion

More formulaic: Iteration is a special case of recursion.

17

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SLIDE 84

Converting Iteration to Recursion

More formulaic: Iteration is a special case of recursion. Idea: The state of an iteration can be passed as arguments.

17

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SLIDE 85

Converting Iteration to Recursion

More formulaic: Iteration is a special case of recursion. Idea: The state of an iteration can be passed as arguments. def sum_digits_iter(n): digit_sum = 0 while n > 0: n, last = split(n) digit_sum = digit_sum + last return digit_sum

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SLIDE 86

Converting Iteration to Recursion

More formulaic: Iteration is a special case of recursion. Idea: The state of an iteration can be passed as arguments. def sum_digits_iter(n): digit_sum = 0 while n > 0: n, last = split(n) digit_sum = digit_sum + last return digit_sum def sum_digits_rec(n, digit_sum): if n == 0: return digit_sum else: n, last = split(n) return sum_digits_rec(n, digit_sum + last)

17

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SLIDE 87

Converting Iteration to Recursion

More formulaic: Iteration is a special case of recursion. Idea: The state of an iteration can be passed as arguments. def sum_digits_iter(n): digit_sum = 0 while n > 0: n, last = split(n) digit_sum = digit_sum + last return digit_sum def sum_digits_rec(n, digit_sum): if n == 0: return digit_sum else: n, last = split(n) return sum_digits_rec(n, digit_sum + last) Updates via assignment become...

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SLIDE 88

Converting Iteration to Recursion

More formulaic: Iteration is a special case of recursion. Idea: The state of an iteration can be passed as arguments. def sum_digits_iter(n): digit_sum = 0 while n > 0: n, last = split(n) digit_sum = digit_sum + last return digit_sum def sum_digits_rec(n, digit_sum): if n == 0: return digit_sum else: n, last = split(n) return sum_digits_rec(n, digit_sum + last) Updates via assignment become... ...arguments to a recursive call

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