Recursion

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Recursion
To Iterate is Human, to Recurse, Divine L.
Peter Deutsch
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Functions reminder
return-type name(arg_type1 arg_name1, arg_type2
arg_name2, ) function body return value

• a group of variables and statements that is
  assigned a name
• a sub-program
• A function can call other functions.

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Return Statement - reminder

• Return causes the execution of the function to
  terminate and returns a value to the calling
  function.
• The type of the value returned must be the same
  as the return-type defined for the function

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Scope of variables - reminder

• A variable declared within a function is
  unrelated to variables declared elsewhere
• A function cannot access variables that are
  declared in other functions

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Func. Declaration - reminder

• We can call a function from the point in the file
  in which the function has been declared, until
  the end of the file.

return-type name(arg_type1 arg_name1, arg_type2
arg_name2, )
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Recursive Function

• A function defined in terms of itself is called a
  recursive function.

return_value rec_func(type arg1, type arg2, )
rec_func()
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Factorial

• As we saw, n! 123 (n-1)n
• Thus, we can also define factorial the following
  way
• 0! 1
• n! n(n-1)! for ngt0

n
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A recursive definition

• C functions can also call themselves!
• However, not with the same parameters (why?)
• Some functions can be defined using smaller
  occurrences of themselves.
• Such a definition is called a recursive
  definition of a function.
• Every recursive function has a boundary
  condition. The function stops calling itself
  when it is satisfied.
• Why is this necessary?

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Example - factorial

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

• int fact_itr(int n)
• int fact 1 while (n gt 1)
• fact n
• --n return fact

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Recursive factorial step by step
n ? 4return ? ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? ?
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
n ? 2return ? ?
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
n ? 2return ? 2 ?
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
n ? 2return ? 2 ?
n ? 1return ? ?
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
n ? 2return ? 2 ?
n ? 1return ? 1 ?
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Recursive factorial step by step

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

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Recursive factorial step by step

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
n ? 2return ? 2 ?
n ? 1return ? 1 1
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 ?
n ? 2return ? 2 1
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Recursive factorial step by step
n ? 4return ? 4 ?

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

n ? 3return ? 3 2
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Recursive factorial step by step
n ? 4return ? 4 6

• int fact_rec(int n)
• if (n 0)
• return 1
• return n fact_rec(n-1)

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Another Example

• void print_nums(int num)
• if (num 0)
• return
• printf("d ", num)
• print_nums(num - 1)
• printf("d ", num)

3 2 1 1 2 3
printcall funcprint
3
printcall funcprint
2
printcall funcprint
1
return
0
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What does it do?

• void foo()
• int num
• scanf("d", num)
• if (num lt 0)
• return
• foo()
• printf("d ", num)

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Reverse Print

• void reverse_print()
• int num
• scanf("d", num)
• if (num lt 0)
• return
• reverse_print()
• printf("d ", num)

Reads an unbounded series of numbers from the
user and prints them in reverse order.
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Another example - power

• Xy xxx
• Recursive definitions (assume non-negative y)
  1, y 0 Xy Xy-1, y
  odd (Xy/2)2, y even

y times
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rec_pow

• int rec_pow(int x, int y)
• if (y 0)
• return 1
• if (y 2 0)
• return square(rec_pow(x, y / 2))
• else
• return x rec_pow(x, y - 1)

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The Three Rules of Recursion

• Know when to stop.
• Decide how to take one step.
• Break the problem down into that step plus a
  smaller problem.

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Exercise

• Write a program that receives two non-negative
  integers and computes their product recursively.
• Hint Notice that the product ab is actually
  aaa (b times).

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Solution

• int rec_mul(int a, int b)
• / base condition a 0 0 /
• if (b 0)
• return 0
• / save a and call recursively /
• return a rec_mul(a, b-1)

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Exercise

• int sum_digits(int n)
• int sum 0
• while (n gt 0)
• sum n10 n n/10
• return sum

• Given the following iterative version of
  sum-of-digits calculation
• Find the recursive definition of this function
• (dont forget the base case!)

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Solution

• int sum_digit_rec(int n)
• return (n 0) ? 0 n 10
  sum_digit_rec(n / 10)

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More uses

• Recursion is a general approach to programming
  functions
• Its uses are not confined to calculating
  mathematical expressions!
• For example max_rec.c

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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max_rec step by step

• int rec_max(int arr , int size)
• int rest
• if (size 1)
• return arr0
• else
• rest rec_max(arr1, size-1)
• if (arr0 gt rest)
• return arr0
• else
• return rest

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What does it do?

• char rec_func(char str, char c)
• if (str '\0')
• return NULL
• if (str c)
• return str
• return rec_func(str, c)

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Solution

• A recursive implementation of strchr
• See strchr_rec.c

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Exercise

• Write a recursive implementation of strcmp
• Input two strings
• Output 0 if both are equal, 1 if not
• Write a program that accepts two strings from the
  user and checks whether they are equal

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Solution

• int rec_strcmp(char s1, char s2)
• if (s1 ! s2)
• return s1 - s2
• if (s1 '\0') return 0
• return rec_strcmp(s1, s2)

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Towers of Hanoi

• Goal
• Move the entire tower to the dst peg.
• Constraints
• Only move one disk at a time.
• Never place a larger disk on a smaller one.

src
dst
aux
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Towers of Hanoi
Step 1 Move n 1 disks from src to aux using
dst.
src
dst
aux
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Towers of Hanoi
Step 1 Move a single disk from src to dst
src
dst
aux
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Towers of Hanoi
Step 3 Move n 1 disks from aux to dst using src
src
dst
aux
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Towers of Hanoi - C

• void hanoi(int n, char src, char
  aux, char dst)
• if (n 1)
• printf("move disk from s to s\n", src,
  dst)
• else
• hanoi(n - 1, src, dst, aux)
• hanoi(1, src, NULL, dst)
• hanoi(n - 1, aux, src, dst)

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