1. Module 6: Recursion Ron K. Cytron*
Department of Computer Science and Engineering
Washington University in Saint Louis
Thanks to Alan Waldman
for comments that improved these slides *
Title slide
Prepared for 2u
Semester Online
Copyright Ron K. Cytron 2013

2. The material we study here involves no new syntax
6.0 Introduction
Monolog 0
The material we study here involves no new syntax
Instead, we will be analyzing and writing methods, such as we studied in Module 5
We have seen methods calling other methods
Compute the distance between two points
We need the sqrt method
Multiplication (pedagogical example)
mpy(c,d) calls add(a,b)
Now we examine methods that call themselves
Seems impossible to solve problem P using P
But it works under the right circumstances
And is a very powerful technique

3. Why study recursion? Could we live without knowing recursion?
6.0 Introduction
Monolog 0
Why study recursion? Could we live without knowing recursion?
Yes: with arrays, iteration, and choice, we know enough to solve any problem that can be solved on a computer.
Mechanisms that allow us to write "any program" are called Turing-complete
However, recursion is a powerful technique for specifying computation
Mathematically satisfying
And some computations are more naturally specified using recursion

4. Nature offers many examples of recursion
6.0 Introduction
Nature offers many examples of recursion
A tree grows its branches by applying the same pattern "in the small" as it does "in the large
Shapes manipulated recursively yield interesting results
From Sedgewick's book

5. Reproductive Parts M. C. Escher, 1948
6.0 Introduction
Nature offers many examples of recursion
But recursion sometimes seems to do the impossible
5 minutes
Reproductive Parts M. C. Escher, 1948
End of Monologue

6. Some formula are explicitly recursive
6.1 Explicit recursion
Oyster 1
Some formula are explicitly recursive
factorial(n) = n * factorial(n-1), if n > 0
factorial(n) = 1, otherwise

7. Some formula are explicitly recursive
6.1 Explicit recursion
Some formula are explicitly recursive
factorial(n) = n * factorial(n-1), if n > 0
factorial(n) = 1, otherwise
This works because a larger problem
factorial(n)

8. Some formula are explicitly recursive
6.1 Explicit recursion
Some formula are explicitly recursive
factorial(n) = n * factorial(n-1), if n > 0
factorial(n) = 1, otherwise
This works because a larger problem
factorial(n)
is phrased in terms of a smaller problem
factorial(n-1)

9. Some formula are explicitly recursive
6.1 Explicit recursion
Some formula are explicitly recursive
factorial(n) = n * factorial(n-1), if n > 0
factorial(n) = 1, otherwise
This works because a larger problem
factorial(n)
is phrased in terms of a smaller problem
factorial(n-1)
and because the computation bottoms out with a base case
factorial(0) = 1

10. Some formula are explicitly recursive
6.1 Explicit recursion
Some formula are explicitly recursive
factorial(n) = n * factorial(n-1), if n > 0
factorial(n) = 1, otherwise
This is the standard way such formulae are defined in math

11. Some formula are explicitly recursive
6.1 Explicit recursion
Some formula are explicitly recursive
factorial(n) = 1, if n <= 0
factorial(n) = n * factorial(n-1), otherwise
Rewrote the definition to put the base case first
Common in computer science

12. Some formula are explicitly recursive
6.1 Explicit recursion
Some formula are explicitly recursive
factorial(n) = 1, if n <= 0
factorial(n) = n * factorial(n-1), otherwise
These are simply transcribed into recursive functions
public static int factorial(int n) {
if (n <= 0)
return 1;
else
return n * factorial(n-1); }

13. 6.1 Explicit recursion
This looks like ordinary code using methods
The only difference is that the method calls itself
Let's look at the structure of a recursive method in more detail
public static int factorial(int n) {
if (n <= 0)
return 1;
else
return n * factorial(n-1); }

14. 6.1 Explicit recursion
public static int factorial(int n) { if (n <= 0) return 1; else return n * factorial(n-1); }
recursive call

15. 6.1 Explicit recursion
public static int factorial(int n) { if (n <= 0) return 1; else return n * factorial(n-1); }
base case
5 minutes
End of Oyster

16. Point students to their repo for the upcoming exercise
6.2a Exercise
BLT 2a public
Video intro
Implement the factorial function
Write unit tests
Debugger and tracing through the code
Point students to their repo for the upcoming exercise
Video intro: 5 minutes, using screenflow

17. 6.2a Exercise Question card Implement and test You do the same for:
sum(n) = sum(n-1) + n, if n > 0
sum(n) = 0, otherwise
In your code, identify (make comments on)
Recursive call
Base case
Implement and test
This could take students 10 minutes to complete

18. 6.2 Exercise Video response Show solution End of BLT
Response: 6 minutes, using screenflow
End of BLT

19. Same setup as previous BLT so this can be a short video intro
6.2b Exercise
BLT 2b public
Same setup as previous BLT so this can be a short video intro
Insist students look at each others' work before continuing (is this possible?)
1 minute intro, no screenflow needed

20. Under what circumstances does your method work?
6.2b Exercise
Question card
Now do the same for:
add(x,y) = x, if y == 0
add(x,y) = add(x+1, y-1), otherwise
In your code, identify (make comments on)
Recursive call
Base case
Implement and test
Under what circumstances does your method work?
How could you generalize it to work for any x and y?
Some 15 minutes for students to do this work

21. 6.2b Exercise Video response Show solution End of BLT
Response: 6 minutes, using screenflow
End of BLT

22. 6.3 Finding recursive substructure
Oyster 3
Sometimes, we have to find the recursion in a problem
It's not explicitly stated
This is harder, but more like solving a puzzle
Requires a bit of faith and vision
Trust that a method does what we want it to do
See the problem in terms of its simplest subproblem

23. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
sum
… +

24. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
sum
… +

25. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
sum(x) = … + (x-1) + x
sum
… +

26. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
sum(x) = … + (x-1) + x
sum(x+9) = … + x + (x+1) + … + (x+8) + (x+9)
sum
… +

27. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
Where in this expression do we see a smaller example of the expression itself?
sum
… +

28. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
Where in this expression do we see a smaller example of the expression itself?
sum
… +

29. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
sum(3) =
Where in this expression do we see a smaller example of the expression itself?
sum
… +

30. 6.3 Finding recursive substructure
Example
sum(n) = … + n
Think of the sum(n) function as a generator of the expression you see on the right
Believe that sum(n) really generates such an expression
sum(4) =
sum(3) =
sum(4) = sum(3)
Where in this expression do we see a smaller example of the expression itself?
sum
… +

31. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n

32. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = … (n-1) n

33. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = … (n-1) n
Aha, the boxed portion also looks like sum(something), but sum of what?

34. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = … (n-1) n
sum(n-1)
if we believe sum(x) really generates
… + x

35. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = sum(n-1) n
sum(n-1)
if we believe sum(x) really generates
… + x

36. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = sum(n-1) n
Now we have an explicitly recursive formula.

37. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = sum(n-1) n
sum(?) = ?
Now we have an explicitly recursive formula.
But it is missing a base case.
What is the smallest example of sum(n)?

38. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
sum(n) = sum(n-1) n
sum(0) = 0
Now we have an explicitly recursive formula.
But it is missing a base case.
What is the smallest example of sum(n)?

39. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
Leads to
sum(n) = 0, if n == 0
sum(n) = sum(n-1) + n, otherwise

40. 6.3 Finding recursive substructure
Where is the recursive substructure?
sum(n) = … + n
Leads to
sum(n) = 0, if n == 0
sum(n) = sum(n-1) + n, otherwise
private static int sum(int n) {
if (n == 0)
return 0;
else
return sum(n-1) + n; }

41. 6.3 Finding recursive substructure
Why does the following substructure not work?
sum(n) = … + n
sum(n) = … (n-1) n

42. 6.3 Finding recursive substructure
Why does the following substructure not work?
sum(n) = … + n
sum(n) = … (n-1) n
Boxed expression is
sum(n) - 0

43. 6.3 Finding recursive substructure
Why does the following substructure not work?
sum(n) = … + n
sum(n) = … (n-1) n
Leads to:
sum(n) = 0 + sum(n) - 0

44. 6.3 Finding recursive substructure
Why does the following substructure not work?
sum(n) = … + n
sum(n) = … (n-1) n
Leads to:
sum(n) = 0 + sum(n) – 0
This is true, mathematically

45. 6.3 Finding recursive substructure
Why does the following substructure not work?
sum(n) = … + n
sum(n) = … (n-1) n
Leads to:
sum(n) = 0 + sum(n) – 0
This is true, mathematically
But it does not lead to a simpler use of the sum function

46. 6.3 Finding recursive substructure
Why does the following substructure not work?
sum(n) = … + n
sum(n) = … (n-1) n
Leads to:
sum(n) = 0 + sum(n) – 0
This is true, mathematically
But it does not lead to a simpler use of the sum function

47. 6.3 Finding recursive substructure
To work, we must find a smaller instance of the function
One that seems easier to compute
One that points us toward the base case
sum(n) = … + n
sum(n) = … (n-1) n
15 minutes
End of Oyster

48. 6.3a Visual recursion and base cases
BLT 3a public
Video intro: about 3 minutes
Amazing hand-drawn image in NY Times by Serkan Ozkaya
[Serkan Ozkaya] simply wanted to draw and see printed a faithful copy of all the type and pictures planned for a broadsheet page of this newspaper: this very page you are reading right now, which shows his version of the page you are reading right now, which shows his version of his version of the page you are reading right now, which...
Each box is smaller than its containing box. Where does this recursion stop?

49. 6.3a Visual recursion and base cases
Before we understood human reproduction, it was thought
that a woman was born with all of the complete humans inside of
her that would ever be born

50. 6.3a Visual recursion and base cases
Before we understood human reproduction, it was thought
that a woman was born with all of the complete humans inside of
her that would ever be born

51. 6.3a Visual recursion and base cases
Before we understood human reproduction, it was thought
that a woman was born with all of the complete humans inside of
her that would ever be born. What could possibly go wrong?

52. 6.3a Visual recursion and base cases
Question card
What is wrong with the procreation story?
Write and submit your answer
5 minutes for students to think and write this up

53. 6.3a Visual recursion and base cases
Question card
Try factorial without its base case
public static int factorial(int n) {
return n * factorial(n-1); }
What do you see when you run this on factorial(2)?
5 more minutes for students to think and write this up

54. 6.3a Visual recursion and base cases
What could possibly go wrong?
Response: presumes an infinite amount of matter
5 minutes for response, including some 2 minutes of screenflow

55. 6.3a Visual recursion and base cases
For factorial
We see stack overflow
We could make the stack bigger
But it would still overflow
Recursion without base cases
leads to stack overflow
This continues the response, and the screenflow is for this part
End of BLT

56. For formulae, find substructure
6.3b Roundtable
Roundtable 3b
For formulae, find substructure
sum(n) = n + (n-1) + (n-2) + … + 0
This is backwards from our earlier example
Multiplication
mpy(x,y) = x + x x (y times)
Do some pictures, finding substructure
Circles
Drawing a line
Sierpinski
square snowflake from intro
graph paper
Why not ½ horizontally or vertically?
10 minutes, not much code here, just formulae and drawings. We want to refer to and point at the drawings and formula, perhaps on a shared screen?

57. Let's try to generate graph paper recursively
6.4 Example
Oyster 4
Let's try to generate graph paper recursively

58. 6.4 Example
This is our goal
but how do we
see this
recursively?

59. 6.4 Example
This is our goal
but how do we
see this
recursively?

60. 6.4 Example
size
(llx,lly)

61. 6.4 Example
size
(llx, lly+size/2)
(llx,lly)

62. 6.4 Example
size
(llx+size, lly+size/2)
(llx, lly+size/2)
(llx,lly)

63. 6.4 Example size (llx+size, lly+size/2) (llx, lly+size/2) (llx,lly)

64. 6.4 Example size (llx+size/2, lly+size) (llx+size, lly+size/2)
(llx, lly+size/2)
(llx,lly)
(llx+size/2, lly)

65. 6.4 Example
size ul ul ur
(ulx, uly) ll lr
(llx,lly)

66. 6.4 Example
size ur ul ur
(urx, ury) ll lr
(llx,lly)

67. Example
size ul ur ll ll lr
(llx, lly)

68. Example
size ul ur lr ll lr
(lrx, lry)
(llx,lly)

69. 6.4 Example
size
size/2 ul ul ur
(llx, lly+size/2) ll lr
(llx,lly)

70. (llx+size/2,lly+size/2)
6.4 Example
size
size/2 ur ul ur
(llx+size/2,lly+size/2) ll lr
(llx,lly)

71. Example
size ul ur
size/2 ll ll lr
(llx, lly)

72. Example size ul ur size/2 lr ll lr (llx+size/2, lly) (llx,lly)
6 minutes
(llx+size/2, lly)
(llx,lly)
End of Oyster

73. 6.5 Exercise Video intro Review slides and coordinates BLT 5 public
Video intro is about two minutes

74. 6.5 Exercise Question card Think about Implement graph paper problem
Simplest substructure
Base case
This could take students 15 minutes to complete

75. 6.5 Exercise Video response Show solution End of BLT
Video about 6 minutes, using screenflow
End of BLT

76. 6.6 There is no 6.6

77. How do we reason about the recursive behavior of methods?
6.7 Substitution model
Oyster 7
How do we reason about the recursive behavior of methods?
If they are functional
meaning, they have no side effects
Then we can substitute recursive calls mathematically
wherever we see
foo(x)
we substitute the text of foo using x as its input
Let's look at factorial again….

78. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) =

79. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 *

80. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 *

81. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1)

82. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1)

83. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1)

84. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1) = 3 * 2 * 1 * fact(0)

85. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1) = 3 * 2 * 1 * fact(0)

86. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1) = 3 * 2 * 1 * fact(0) = 3 * 2 * 1 * 1

87. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1) = 3 * 2 * 1 * fact(0) = 3 * 2 * 1 * 1 = 3 * 2 * 1

88. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1) = 3 * 2 * 1 * fact(0) = 3 * 2 * 1 * 1 = 3 * 2 * 1 = 3 * 2

89. Substitution model for recursive evaluation
public static int fact(n) { if (n==0) return 1; // I tend to put base case first else return n * fact(n-1); } fact(3) = 3 * fact(3-1) = 3 * fact(2) = 3 * 2 * fact(2-1) = 3 * 2 * fact(1) = 3 * 2 * 1 * fact(1-1) = 3 * 2 * 1 * fact(0) = 3 * 2 * 1 * 1 = 3 * 2 * 1 = 3 * 2 = 6

90. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – …. = 0 ?
No, must keep computation in the boxes

91. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – …. = 0 ?
No, must keep computation in the boxes
Each box deserves a pair of parentheses

92. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1)

93. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0)

94. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0
mys(0) = 0

95. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0 = 3 – 2 – 1 - 0
mys(0) = 0
mys(1) = 1-0 = 1

96. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0 = 3 – 2 – 1
mys(0) = 0
mys(1) = 1-0 = 1

97. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0 = 3 – 2 – 1 = 3 – 2 - 1
mys(0) = 0
mys(1) = 1-0 = 1
mys(2) = 2-1 = 1

98. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0 = 3 – 2 – 1 = 3 – 1
mys(0) = 0
mys(1) = 1-0 = 1
mys(2) = 2-1 = 1

99. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0 = 3 – 2 – 1 = 3 – 1 = 3 - 1
mys(0) = 0
mys(1) = 1-0 = 1
mys(2) = 2-1 = 1

100. Substitution model for recursive evaluation
public static int mys(n) { if (n==0) return 0; // I tend to put base case first else return n – mys(n-1); } mys(3) = 3 – mys(2) = 3 – 2 – mys(1) = 3 – 2 – 1 – mys(0) = 3 – 2 – 1 – 0 = 3 – 2 – 1 = 3 – 1 = 2
mys(0) = 0
12 minutes
mys(1) = 1-0 = 1
mys(2) = 2-1 = 1
End of Oyster

101. 6.7b Practice doing substitution
BLT 7b public
Let's practice substitution
Always write = down the page
fact(3) = 3 * fact(2) =
etc
2 minute intro

102. 6.7b Practice doing substitution
Question card
For sum(n)
sum(4)
For add(x,y) now show substitution for
add(10,3)
20 minutes for students to work on this

103. 6.7b Practice doing substitution
Video response
For sum(n)
sum(4)
For add(x,y) now show substitution for
add(10,3)
Response video 5 minutes, on a board or paper, no screenflow
End of BLT

104. Use the debugger to trace recursive calls
6.7c Tracing recursion
Roundtable
Use the debugger to trace recursive calls
Find the code in your repository
2 minute intro

105. Go over the following with each of the 3 students in roundtable:
6.7c Tracing recursion
Go over the following with each of the 3 students in roundtable:
For sum(n)
sum(4)
For add(x,y) now show substitution for
add(10,3)
For mys(n)
mys(3)
15 total minutes of roundtable. The code would already be prepared, and the student would trace through it with me watching and commenting.
End of Roundtable

106. Recursion is a powerful technique
6.8 Conclusion
Monolog 8
Recursion is a powerful technique
Nice results with relatively little coding
Code often directly reflects the specified computation
Ingredients of recursion
Base case
Recursive call
Sometimes we have to find the substructure
Best to find the next smallest subcase of the larger problem
Substitution model helps us reason about recursion
2 minutes
End of Monologue