Lecture 10

Iterative Recursion - A Slightly Different Flavour

Basic template for recursion

• Recursion is about
• Solving a complicated problem
• by solving a simpler version of the problem

​

• But you have to stop sometime
• Stop when you get to “easy” problem

​

• You need to write code for
• Recognizing easy cases
• Solving easy (base) cases
• Simplifying the problem (get one step closer to the base/easy cases)
• Fixing the simplified solution into a solution to the full problem

​

(define function

(lambda (args)� (if easy-case?� solve-easy-case� (fix-solution� (function simplified))))

Special kinds of recursion

• Multiple kinds of easy cases
• Iterative recursion
• Fix-up part is blank
• Instead update accumulator in recursive step
• Tree recursion
• Function calls itself more than once
• Infinite recursion (usually bad)
• No check for easy case
• Broken check for easy case
• Broken code for simplifying the args

(define function

(lambda (args)� (if easy-case?� solve-easy-case� (fix-solution� (function simplified))))

Iterative recursion

Summing a list using recursion

We can add up the elements of a list by

• Removing the first element
• Adding the remaining elements (recursive step)
• Adding the first element into the total

;; sum-list:

;; (listof number) -> number

;; Total of elements of a list

(define sum-list

(lambda (sum-list)

(if (empty? list)

0

(+ (first list)

(sum-list (rest list))))))

Is that the only way to do it?

• This code adds the first element after recursing
• Is there a way to add before recursing?

;; sum-list:

;; (listof number) -> number

;; Total of elements of a list

(define sum-list

(lambda (sum-list)

(if (empty? list)

0

(+ (first list)

(sum-list (rest list))))))

Keeping a partial sum (an accumulator)

• We just keep track of the running total
• That ends up having to be an additional argument So we call it by saying �(sum-list (list 1 2 3) 0)
• Saves the computer having to remember to do the addition afterward

;; sum-list:

;; (listof number) -> number

;; Sum of all numbers in list

(define sum-list

(lambda (a-list)

(if (empty? a-list)

0

(+ (first a-list)

(sum-list (rest a-list))))))

​

;; sum-list:

;; (listof number) number -> number

;; Sum of all numbers in list plus partial-sum

(define sum-list

(lambda (a-list partial-sum)

(if (empty? a-list)

partial-sum

(sum-list (rest a-list)

(+ (first a-list)

partial-sum)))))

​

Why it matters

Adding last:

• (sum-list (list 1 2 3 4))
• (+ 1 (sum-list '(2 3 4)))
• (+ 1 (+ 2 (sum-list '(3 4))))
• (+ 1 (+ 2 (+ 3 (sum-list '(4)))))
• (+ 1 (+ 2 (+ 3 (+ 4 (sum-list

'())))))

• (+ 1 (+ 2 (+ 3 (+ 4 0))))
• (+ 1 (+ 2 (+ 3 4)))
• (+ 1 (+ 2 7))
• (+ 1 9)
• 10

Adding first:

• (sum-list '(1 2 3 4) 0)
• (sum-list '(2 3 4) (+ 1 0))
• (sum-list '(2 3 4) 1)
• (sum-list '(3 4) (+ 2 1))
• (sum-list '(3 4) 3)
• (sum-list '(4) (+ 3 3))
• (sum-list '(4) 6)
• (sum-list '() (+ 4 6))
• (sum-list '() 10)
• 10

​

Holding + until last requires more stack space to keep track pending operations.

Iteration and recursion

• When a procedure calls itself, it’s called a recursive function.
• When the procedure returns the value of the call directly, without a fixup step, it’s called iterative recursion
• Also known as tail recursion (or looping)
• Doesn’t require the computer to remember pending operations
• Less memory to execute
• CS 211 Preview: iterative recursion can re-use the current stack frame, allowing the recursion to be compiled as a while loop
• Iterative recursion generally requires that an accumulator argument hold the intermediate result
• The base case is usually just to return that argument

How do we write foldl?

• We can use the same trick to write foldl!

;; sum-list:

;; (listof number) number -> number

;; Sum of all numbers in list plus partial-sum

(define sum-list

(lambda (a-list partial-sum)

(if (empty? a-list)

partial-sum

(sum-list (rest a-list)

(+ (first a-list)

partial-sum)))))

How do we write foldl?

• We can use the same trick to write foldl!
• What we’ve been calling the start argument before
• Is really just the partial result (an accumulator)

;; foldl:

;; (X X -> X) X (listof X) -> X

;; Uses func to collapse all elements of list

;; plus partial-sum

(define foldl

(lambda (func partial-result a-list)

(if (empty? a-list)

partial-result

(foldl func

(func (first a-list)

partial-result)

(rest a-list))))

​

​

Helper procedures

Example

• Write length iteratively (it requires two functions)

​

;; length: (listof X) -> number

;; Number of elements in list

​

Example

• Write length iteratively (you’ll need to write two functions)�

;; length: (listof X) -> number

;; Number of elements in list

(define length

(lambda (a-list)

(length-loop a-list 0)))

​

;; length-loop: (listof X) number -> number

;; Number of elements in list plus length-so-far

(define length-loop

(lambda (list-remaining length-so-far)

(if (empty? list-remaining)

length-so-far

(length-loop (rest list-remaining)

(+ length-so-far 1)))))

​

Execution under the substitution model

• (length ‘(1 2 3))
• (length-loop '(1 2 3) 0)
• (if (empty? '(1 2 3))

0

(length-loop (rest '(1 2 3))

(+ 0 1)))

• (if false

0

(length-loop (rest '(1 2 3))

(+ 0 1)))

• (length-loop (rest '(1 2 3))

(+ 0 1)))

• (length-loop '(2 3) (+ 0 1))
• (length-loop '(2 3) 1)
• (if (empty? '(2 3))

1

(length-loop (rest '(2 3))

(+ 1 1)))

​

• (if false

1

(length-loop (rest '(2 3))

(+ 1 1)))

• (length-loop (rest '(2 3))

(+ 1 1)))

• (length-loop '(3) (+ 1 1)))
• (length-loop '(3) 2)
• (if (empty? '(3))

2

(length-loop (rest '(3)) (+ 2 1)))

• (if false

2

(length-loop (rest '(3)) (+ 2 1)))

• (length-loop (rest '(3)) (+ 2 1))

​

​

Execution under the substitution model

• (length-loop '() (+ 2 1))
• (length-loop '() 3)
• (if (empty? '())

3

(length-loop (rest '())

(+ 3 1)))

• (if true

3

(length-loop (rest '())

(+ 3 1)))

• 3

​

Making it prettier

• You can place function definitions within other functions
• Those variables are then only available within the procedure
• The expressions within them can use the other variables of the procedure�

;; length: (listof X) -> number

;; Number of elements in list

(define my-length

(lambda (a-list)

;; loop: (listof X) number -> number

;; Number of elements in list plus length-so-far

(local [(define loop

(lambda (a-list length-so-far)

(if (empty? a-list)

length-so-far

(loop (rest a-list)

(+ length-so-far 1)))))]

(loop a-list 0))))

Iterative Recursion

• Instead of waiting until the end to "combine" all the simpler answers
• We could instead keep track of the "answer so far" in an accumulator
• This is called iterative recursion
• Uses a helper function to maintain the correct number of args.

(define (help args … accum)

(if easy-case?� accum� (help args …

(updater accum …))))

​

(define (func args …) � (help args … start-accum))

Tree recursion

Basic template for recursion

• Recursion is about
• Solving a complicated problem
• by solving a simpler version of the problem

​

• But you have to stop sometime
• Stop when you get to “easy” problem

​

• You need to write code for
• Recognizing easy cases
• Solving easy (base) cases
• Simplifying the problem (get one step closer to the base/easy cases)
• Fixing the simplified solution into a solution to the full problem

​

(define function

(lambda (args)� (if easy-case?� solve-easy-case� (fix-solution� (function simplified))))

Splitting the Problem into Chunks

• You can often simplify the problem by splitting it
• At least one of the splits will then be a recursive call
• Then the fixup step consists of combining the answers to the two problems
• This is called tree recursion

(define (func args …) � (if easy-case?� solve-easy-case� (combine (func one-part)� (func other-part)))))

How google sums a list

• Google uses parallel algorithms do their searches
• Parallel means it runs on several computers at once
• Google splits the list up into chunks and runs each chunk on a different machine
• This is called a parallel-prefix scan operation
• But it really just means folding
• This allows them to use large networks of cheap PCs to do their searches quickly

(define sum-list

(lambda (a-list)

(if (list-fits-on-one-processor? a-list)

(add-it-up a-list)

(+ (sum-list (first-half a-list))

(sum-list (second-half a-list)))))

• A very approximate version of the algorithm Google uses
• Note: the two recursive calls run on different machines

Parallel summation

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list:

Parallel summation

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list: + Sum this list:

Parallel summation

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list: + Sum this list:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list: + Sum this list: Sum this list: + Sum this list:

Parallel summation

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list: + Sum this list:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

10 + 26 42 + 58

Parallel summation

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Sum this list:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

36 + 100

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

10 + 26 42 + 58

Parallel summation

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

136

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

36 + 100

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

10 + 26 42 + 58

Optional reading

Chapter 1 of Abelson and Sussman

Basic template for ordinary recursion

• Recursion is about
• Solving a complicated problem
• by solving a simpler version of the problem

​

• But you have to stop sometime
• Stop when you get to “easy” problem

​

• You need to write code for
• Recognizing easy cases
• Solving easy (base) cases
• Simplifying the problem (get one step closer to the base/easy cases)
• Fixing the simplified solution into a solution to the full problem

​

(define function

(lambda (args)� (if easy-case?� solve-easy-case� (fix-solution� (function simplified))))

Iterative Recursion

• You can often simplify the problem by splitting it
• Then the fixup step consists of combining the answers to the two problems
• This is called tree recursion

(define (func args …) � (if easy-case?� solve-easy-case� (combine (func one-part)� (func other-part)))))

Tree Recursion

• Instead of waiting until the end to "combine" all the simpler answers
• We could instead keep track of the "answer so far" in an accumulator
• This is called iterative recursion
• Uses a helper function to maintain the correct number of args.

(define (help args … accum)

(if easy-case?� accum� (help args …

(updater accum …))))

​

(define (func args …) � (help args … start-accum))