1. CMSC 2021 Recursion Recursive Definition – one that defines something in terms of itself Recursion – A technique that allows us to break down a problem in to one or more subproblems that are similar in form to the original problem. Either the problem or the associated data can be recursive in nature

2. CMSC 2022 Iterative Factorial Iterative definition N! = N * (N-1) * (N-2) * … * 3 * 2 * 1 int factorial (int n) { int i, result; result = 1; for (i = 1; i <= n; i++) { result = result * i; } return ( result ); }

3. CMSC 2023 Recursive Factorial Recursive N! definition N! = 1 if N = 0 = N * (N-1)! Otherwise int factorial ( int n) { if (n == 0) return (1); return (n * factorial(n - 1) ); }

4. CMSC 2024 Iterative vs Recursive factorial Iterative 1.2 local variables 2.3 statements 3.Saves solution in an intermediate variable Recursive 1. No local variables 2. One statement 3. Returns result in single expression Recursion simplifies factorial by making the computer do the work. What work and how is it done?

5. CMSC 2025 How does it work? Each recursive call creates an “activation record” which contains local variables and parameters return address and is saved on the stack.

6. CMSC 2026 Iterative Power Iterative definition X N = X * X *X ….. * X (N factors) double power(double number, int exponent) { double p = 1; for (int i = 1; i <= exponent; i++) p = p * p; return (p); }

7. CMSC 2027 Recursive Power Recursive definition X N = 1 if N = 0 = X * X (N-1) otherwise double power(double number, int exponent) // Precondition: exponent >= 0 { if (exponent == 0) return (1); return (number * power(number, exponent-1)); }

8. CMSC 2028 Recursive String Reversal void RString (char *s, int len) { if (len <= 0) return; RString (s+1, len-1); cout << *s ; }

9. CMSC 2029 Recursive Data Structures Array -- a set of data elements store in contiguous memory cells Array -- zero elements OR a single data element OR an array followed by a single element

10. CMSC 20210 Iteratively Sum an Array int SumArray (int a[ ], int size) { int j, sum = 0; for ( j = 0; j < size; j++ ) sum += a[ j ]; return sum; }

11. CMSC 20211 Recursively Sum an Array int SumArray ( int a [ ], int size) { if (size == 0) return 0; else return array[ size – 1] + SumArray (a, size – 1); }

12. CMSC 20212 Approach to Writing Recursive Functions Using “factorial” as an example, 1.Write the function header so that you are sure what the function will do and how it will be called. [For factorial, we want to send in the integer for which we want to compute the factorial. And we want to receive the integer result back.] int factorial(int n)

13. CMSC 20213 Approach (cont’d) 2. Decompose the problem into subproblems (if necessary). For factorial, we must compute (n-1)! 3. Write recursive calls to solve those subproblems whose form is similar to that of the original problem. There is a recursive call to write: factorial(n-1) ; ]

14. CMSC 20214 Approach (cont’d) 4. Write code to combine, augment, or modify the results of the recursive call(s) if necessary to construct the desired return value or create the desired side effects. When a factorial has been computed, the result must be multiplied by the current value of the number for which the factorial was taken: return (n * factorial(n-1));

15. CMSC 20215 Approach (cont’d) 5. Write base case(s) to handle any situations that are not handled properly by the recursive portion of the program. The base case for factorial is that if n=0, the factorial is 1: if (n == 0) return(1);

16. CMSC 20216 Ensuring that Recursion Works Using “factorial” as an example, 1. A recursive subprogram must have at least one base case and one recursive case (it's OK to have more than one base case and/or more than one recursive case). The first condition is met, because if (n==0) return 1 is a base case, while the "else" part includes a recursive call (factorial(n-1)).

17. CMSC 20217 Does It Work? (cont’d) 2. The test for the base case must execute before the recursive call. If we reach the recursive call, we must have already evaluated if (n==0); this is the base case test, so this criterion is met.

18. CMSC 20218 Does It Work? (cont’d) 3. The problem must be broken down in such a way that the recursive call is closer to the base case than the top-level call. The recursive call is factorial(n-1). The argument to the recursive call is one less than the argument to the top-level call to factorial. It is, therefore, “closer” to the base case of n=0.

19. CMSC 20219 Does It Work? (cont’d) 4. The recursive call must not skip over the base case. Because n is an integer, and the recursive call reduces n by just one, it is not possible to skip over the base case.

20. CMSC 20220 Does It Work? (cont’d) 5. The non-recursive portions of the subprogram must operate correctly. Assuming that the recursive call works properly, we must now verify that the rest of the code works properly. We can do this by comparing the code with our definition of factorial. This definition says that if n=0 then n! is one. Our function correctly returns 1 when n is 0. If n is not 0, the definition says that we should return (n-1)! * n. The recursive call (which we now assume to work properly) returns the value of n-1 factorial, which is then multiplied by n. Thus, the non--recursive portions of the function behave as required.

21. CMSC 20221 Recursion vs. Iteration Recursion  Usually less code – algorithm can be easier to follow (Towers of Hanoi, binary tree traversals, flood fill)  Some mathematical and other algorithms are naturally recursive (factorial, summation, series expansions, recursive data structure algorithms)

22. CMSC 20222 Recursion vs. Iteration (cont’d) Iteration  Use when the algorithms are easier to write, understand, and modify (especially for beginning programmers)  Generally, runs faster because there is no stack I/O  Sometimes are forced to use iteration because stack cannot handle enough activation records (power(2, 5000))

23. CMSC 20223 Direct Recursion A C function is directly recursive if it contains an explicit call to itself Int foo (int x) { if ( x <= 0 ) return x; return foo ( x – 1); }

24. CMSC 20224 Indirect Recursion A C function foo is indirectly recursive if it contains a call to another function that ultimately calls foo. int foo (int x){ if (x <= 0) return x; return bar(x); } int bar (int y) { return foo (y – 1); }

25. CMSC 20225 Tail Recursion A recursive function is said to be tail recursive if there is no pending operations performed on return from a recursive call Tail recursive functions are often said to “return the value of the last recursive call as the value of the function”

26. CMSC 20226 factorial is non-tail recursive int factorial (int n) { if (n == 0) return 1; return n * factorial(n – 1); } Note the “pending operation” (namely the multiplication) to be performed on return from each recursive call

27. CMSC 20227 Tail recursive factorial int fact_aux (int n, int result) { if (n == 1) return result; return fact_aux (n – 1, n * result); } factorial (int n) { return fact_aux (n, 1); }

28. CMSC 20228 Linear Recursion A recursive function is said to be linearly recursive when no pending operation invokes another recursive call to the function. int factorial (int n) { if (n == 0) return 1; return n * factorial ( n – 1); }

29. CMSC 20229 Linear Recursive Count_42s int count_42s(int array[ ], int n) { if (n == 0) return 0; if (array[ n-1 ] != 42) { return (count_42s( array, n-1 ) ); } return (1 + count_42s( array, n-1) ); }

30. CMSC 20230 Tree Recursion A recursive function is said to be tree recursive (or non- linearly recursive when the pending operation does invoke another recursive call to the function. The Fibonacci function fib provides a classic example of tree recursion.

31. CMSC 20231 Fibonacci Numbers The Fibonacci numbers can be recursively defined by the rule: fib(n) = 0 if n is 0, = 1 if n is 1, = fib(n-1) + fib(n-2) otherwise For example, the first seven Fibonacci numbers are Fib(0) = 0 Fib(1) = 1 Fib(2) = Fib(1) + Fib(0) = 1 Fib(3) = Fib(2) + Fib(1) = 2 Fib(4) = Fib(3) + Fib(2) = 3 Fib(5) = Fib(4) + Fib(3) = 5 Fib(6) = Fib(5) + Fib(4) = 8

32. CMSC 20232 Tree Recursive Fibonacci // Note how the pending operation (the +) // needs a 2 nd recursive call to fib int fib(int n) { if (n == 0) return 0; if (n == 1) return 1; return ( fib ( n – 1 ) + fib ( n – 2 ) ); }

33. CMSC 20233 Tree Recursive Count_42s int count_42s(int array[ ], int low, int high) { if ((low > high) || (low == high && array[low] != 42)) { return 0 ; } if (low == high && array[low] == 42) { return 1; } return (count_42s(array, low, (low + high)/2) + count_42s(array, 1 + (low + high)/2, high)) ); }

34. CMSC 20234 Converting Recursive Functions to Tail Recursion A non-tail recursive function can often be converted to a tail recursive function by means of an “auxiliary” parameter that is used to form the result. The idea is to incorporate the pending operation into the auxiliary parameter in such a way that the recursive call no longer has a pending operations (is tail recursive)

35. CMSC 20235 Tail Recursive Fibonacci int fib_aux ( int n, int next, int result ) { if ( n == 0 ) return result; return ( fib_aux ( n - 1, next + result, next )); } int fib ( int n ) { return fib_aux ( n, 1, 0 ); }

36. CMSC 20236 Converting Tail Recursive Functions to Iterative Let’s assume that any tail recursive function can be expressed in the general form F ( x ) { if ( P( x ) ) return G( x ); return F ( H( x ) ); }

37. CMSC 20237 Tail Recursive to Iterative (cont’d) P( x ) is the base case of F(x) If P(x) is true, then the value of F(x) is the value of some other function, G(x). If P(x) is false, then the value of F(x) is the value of the function F on some other value H(x).

38. CMSC 20238 Tail Recursive to Iterative (cont’d) F( x ) { int temp_x; while ( P(x) is not true ) { temp_x = x; x = H (temp_x); } return G ( x );

39. CMSC 20239 Example – tail recursive factorial int fact_aux (int n, int result) { if (n == 1) return result; return fact_aux (n – 1, n * result) } F = fact_aux x is really two params – n and result value of P(n, result) is value of (n == 1) value of G(n, result) is result value of H(n, result) is (n – 1, n * result)

40. CMSC 20240 Iterative factorial int fact_aux (int n, int result) { int temp_n, temp_result; while ( n != 1 ) { // while P(x) not true temp_n = n;// temp_x = x (1) temp_result = result;// temp_x = x (2) n = temp_n – 1;// H (temp_x) (1) result = temp_n * temp_result;// H (temp_x) (2) } return result;// G (x) }

41. CMSC 20241 Converting Iteration to Tail Recursion Just like it’s possible to convert tail- recursion into iteration, it’s possible to convert any iterative loop into the combination of a recursive function and a call to that function

42. CMSC 20242 A while-loop example The following code fragment uses a while loop to read values into an integer array until EOF occurs, counting the number of integers read. nt nums [MAX]; int count = 0; while (scanf (“%d”, &nums[count]) != EOF) count++;

43. CMSC 20243 While loop example (cont’d) The key to implementing the loop using tail recursion is to determine which variables capture the “state” of the computation and use these variables as parameters to the tail recursive subprogram. count – the number of integers read so far nums -- the array that stores the integers

44. CMSC 20244 While loop example (cont’d) Since we are interested in getting the final value for count, we will write a function that returns count as its result. If the termination condition ( EOF ) is not true, the function increments count, reads a number into nums and calls itself recursively to to input the remainder of the numbers. If the termination condition is true, the function simply returns the final value of count.

45. CMSC 20245 While loop example (cont’d) int read_nums (int count, int nums[ ] ) { if (scanf (“%d”, &nums[count] != EOF)) return read_nums (count + 1, nums); return count; }

46. CMSC 20246 While loop example (cont’d) To call the function, we must pass in the appropriate initial values count = read_nums (0, nums);

47. CMSC 20247 Converting Iteration to Tail recursion Determining the variables used by the loop Writing tail-recursive function with one parameter for each variable Using the loop-termination condition as the base case

48. CMSC 20248 Converting Iteration to Tail recursion (cont’d) Determining how each variable changes from one iteration to the next and handing the sequence of expressions that represent those changes to the recursive call Replacing the iterative loop with a call to the tail recursive function with appropriate initial values

49. CMSC 20249 An exercise for the Student Convert the following for loop that finds the maximum value in an array to be tail recursive. Assume that the function max( ) returns the larger of the two parameters returned to it. int nums[MAX];// filled by some other function int max_num = nums[ 0 ]; for (j = 0; j < MAX; j++) max_num = max (max_num, nums[ j ] );

50. CMSC 20250 Floodfill void Image::floodfill(int x, int y) { if (x = numRows || y = numCols) return; if (a[x][y] == '1') return; if (a[x][y] == '0') a[x][y] = '1'; floodfill(x+1, y); // Go up floodfill(x-1, y); // Go down floodfill(x, y-1); // Go left floodfill(x, y+1); // Go right }