1. Recursion l Powerful Tool l Useful in simplifying a problem (hides details of a problem) l The ability of a function to call itself l A recursive call is a function call in which the called function is the same as the one making the call. l We must avoid making an infinite sequence of function calls (infinite recursion).

2. 2 Recursion: Divide and Conquer Problem Solving Strategy l 1) Divide problem (input instance) into one or more subproblems of exactly the same type as the original problem l 2) Solve each subproblem l 3) Combine solutions of subproblems to obtain a solution for original input l NOTE: Looks like top-down design, except subproblems are the exactly the same type as the original problem

3. Finding a Recursive Solution l Each successive recursive call should bring you closer to a situation in which the answer is known. l A case for which the answer is known (and can be expressed without recursion) is called a base case. (The answer is usually known in the smallest version of the problem that requires the least amount of work, for example print out a list of only 1 element.) l Each recursive algorithm must have at least one base case, as well as the general (recursive) case

4. General format for many recursive functions if (some condition for which answer is known) solution statement// base case else recursive function call // general case SOME EXAMPLES...

5. Writing a recursive function to find n! (factorial) DISCUSSION The function call Factorial(4) should have value 24, because it is 4 * 3 * 2 * 1. For a situation in which the answer is known), the value of 0! is 1. So our base case could be along the lines of if ( number == 0 ) return 1;

6. cont. Now for the general case... The value of Factorial(n) can be written as n * the product of the numbers from (n - 1) to 1, that is, n * (n - 1) *... * 1 or, n * Factorial(n - 1) And notice that the recursive call Factorial(n - 1) gets us “closer” to the base case of Factorial(0).

7. Recursive Solution int Factorial ( int number ) // Pre: number is initialized and number >= 0. { if ( number == 0)// base case return 1 ; else// general case return number + Factorial ( number - 1 ); }

8. Writing a recursive function to find Nth Fibonacci number DISCUSSION Conceived in 13 th century by Italian mathematician to model how fast rabbits could breed http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13, 21, … 1 st number is 0 2 nd number is 1 3 rd number is 2 nd number + 1 st number … nth number is the n-1 st number + n-2 nd number

9. cont. For situation(s) in which the answer is known, the value of 1 st Fibonacci number is 0 while the value of the 2 nd is 1. So our base cases are if ( N == 1 ) return 0; else if (N == 2) return 1; The general case is The value of Nth Fibonacci number can be written as the sum of the n-1 st and n-2 nd Fibonacci numbers or, Fibonacci (N-1) + Fibonnaci (N-2) Notice that the both recursive calls get us “closer” to the base cases of Fibonnaci (1) or Fibonnaci (2).

10. Recursive Solution int Fib ( int N ) // Pre: N is assigned and N >= 0. { if ( N == 1)// base case return 0; else if ( N == 2)// base case return 1; else// general case return Fib (N-1) + Fib ( N – 2 ); }

11. Three-Question Method of Verifying Recursive Functions l Base-Case Question: Is there a nonrecursive way out of the function that works? l Smaller-Caller Question: Does each recursive function call involve a smaller case of the original problem leading to the base case? l General-Case Question: Assuming each recursive call works correctly, does the whole function work correctly?

12. 12 Fibonacci Example l Base cases: N=1 and N=2 n when N=1, returns 0 ….correct by definition n when N=2, returns 1 …. correct by definition l Recursive calls smaller n 1 st passes N-1 and 2 nd passes N-2 …check l If we assume Fib(N-1) and Fib(N-2) give us correct values of the N-1 st and N-2 numbers in the series, and the assignment statement adds them together …this is the definition of the Nth Fibonacci number, so we know the function works for N > 2. l Since the function works for N=1, N=2, N>2, then it works for N >= 1. l Therefore the function works. (Induction Proof)

13. Another example where recursion comes naturally l From mathematics, we know that 2 0 = 1 and 2 5 = 2 * 2 4 l In general, x 0 = 1 and x n = x * x n-1 for integer x, and integer n > 0. l Here we are defining x n recursively, in terms of x n-1

14. // Recursive definition of power function int Power ( int x, int n ) // Pre: n >= 0. x, n are not both zero // Post: Function value = x raised to the power n. { // base case if ( n == 0 ) return 1; // general case else return ( x * Power ( x, n-1 ) ) ; } Of course, an alternative would have been to use looping instead of a recursive call in the function body. 14

15. Recursive function to determine if value is in array PROTOTYPE void DoRetrieveItem (ArrayType info, ItemType & item, bool & found, int start, int end); Already searched Needs to be searched 74 36... 95 info[0] [1] [start] 75 29 47... [length -1] index of current element to examine

16. void RetrieveItem (ArrayType info, ItemType & item, bool & end) // Searches list for element with same key as item // Pre: item’s key is initialized // Post: if key of item exists in an item in the list, item will be set to that element and found will // be set to true, otherwise found will be set to false { //initial call to recursive private member function DoRetrieveItem (info, item, found, 0 ) ; } 16

17. void DoRetrieveItem ( ArrayType info, ItemType & item, bool & found, int start, int end ) // Searches list info for value between positions start and length-1 // Pre: the sublist, list.info[start ].. list.info[ list.length-1] contain values to be searched // Post: if key of item exists in an item in the sublist info[start]... info[ list.end], item will // be set to that element and found will be set to true, otherwise found will be set to false { if (start == end ) // one base case return false ; else if ( info[start] == value ) // another base case { item = info [start]; return true ; } else // general case return DoRetrieveItem (info, item, found, start + 1, end) ; } 17

18. void DoRetrieveItem ( ArrayType info, ItemType & item, bool & found, int start, int length ) // Searches list for value between positions start and end // Pre: the sublist, list.info[start ].. list.info[ list.end] contain values to be searched and is SORTED // Post: if key of item exists in an item in the sublist list.info[start]... list.info[ list.end], item will // be set to that element and found will be set to true, otherwise found will be set to false { int mid; if (start > end) found = false; // base case else { mid = (start + end)/2; if (info[mid] == item ) // base case { item = info[mid]; found = true ; } else if (info[mid] < item ) // general case DoRetrieveItem (info, item, found, start, mid-1); else // general case DoRetrieveItem (info, item, found, mid+1, end); } 18

19. “Why use recursion?” The first couple of examples could have been written without recursion, using iteration instead. The iterative solution uses a loop, and the recursive solution uses an if statement. However, for certain problems the recursive solution is the most natural solution. In a future example, you will see a function to print a stack in reverse. I will use both a loop and a temporary container. It is easier to solve this problem using recursion.