1. Recursion Review: A recursive function calls itself, but with a smaller problem – at least one of the parameters must decrease. The function uses the results of the recursive call to create its result. There must be a base case – when the problem is sufficiently small to be solved directly, or by another method.

2. Recursion - Example Consider the following problem: We want to print out the digits of a number (in decimal) one by one. For example, the number 16425 would be printed out as: 1 6 4 2 5 How do we isolate a single digit of a number? Hint – how do we get the one's place (5 in this example), i.e., what arithmetic operation could we do to get 5 from 16425?

3. Example (cont'd) Now that we can get the one's place (5), how do we print out the rest of the number? What is the rest of the number? Hint – pass the buck.

4. Exercise How could we change the previous code to print out the number in binary?

5. Recursion in General We've seen examples where we code right a program using either iteration or recursive – with the same running times (finding the minimum element, insertion sort, selection sort, mergesort, finding the exponential), and two examples where it's easier to write the program using recursion (quicksort and the Tower of Hanoi).

6. Recursion (cont'd) In theory, we can always use either to solve a problem – but some problems are more naturally solved using recursion, and some with iteration.

7. Bad Recursive Example A classic problem:You have a pair of rabbits. Every two years on the dot, a pair of rabbits produces a new litter of exactly two rabbits. We assume that our rabbits never die. How many pairs of rabbits will we have in, say 6, years.

8. Fibonacci Sequence This problem gives us the Fibonnaci sequence, f 1 = 1, f 2 = 1, f n = f n-1 + f n-2. Since the sequence is defined recursively (the next value of f is defined in terms of two previous values), it would seem natural to write a recursive function to find the nth number in the sequence.

9. Recursive Function int fib(n) { if(n == 1 || n == 2) return 1; else return fib(n-1) + fib(n-2); } What's wrong with this solution?

10. How the Computer Does Recursion Basically, a recursive function is one that plays “Pass the Buck” with itself as its own neighbor. For each instance of the problem, the function keeps a record of who called it (the calling function, and the point in that function when it was called), and the arguments – that is, the problem that it was given. When it passes the buck to its neighbor (really itself) it creates a new record with the new problem, and places it on top of the old record.

11. How it Works (cont'd). When it finishes a problem, it removes the top record, and writes the answer in the record that is now on top – the record with the previous problem which has just been solved. When it gets to the base case, it does not play Pass the Buck, but rather just solves that problem directly (runs the appropriate code) and returns the answer to the calling function.

12. Example - Mergesort Suppose that we use the mergesort function to sort the list {4,2,1,7,6}, that is, a = {4,2,1,7,6}, low = 1, and high = 5. A record is created with the calling function (say main ), the position in the calling function, and the arguments. Since it's not the base case, the mergesort function splits the list into two pieces, and recurses, first posing the problem: a, 1, 3.

13. Example (cont'd) A new record is created with the calling function ( mergesort ), the position in that function (the first recursive call to mergesort ) and the new arguments: a, 1, 3. That record is placed on top of the original record (which does not reappear until the new problem is solved). The mergesort function is called again with the new arguments, a new record is created (if not the base case) and so on...

14. Example (cont'd) When the function finishes, it removes the top record, and records the answer (the return value) in the record that now is on top – the one with the problem that the answer is the solution for.

15. Example (cont'd) Calling function: main, line 3 Arguments: a = {4, 2, 1, 7, 6}, 1, 6 Result: Calling function: mergesort, line 4 Arguments: a = {4, 2, 1, 7, 6}, 1, 3 Result: Calling function: mergesort, line 4 Arguments: a = {4, 2, 1, 7, 6}, 1, 2 Result: Calling Function: mergesort, line 4 Arguments: a = {4, 2, 1, 7, 6}, 1, 1 Result: {4}

16. Calling function: main, line 3 Arguments: a = {4, 2, 1, 7, 6}, 1, 6 Result: Calling function: mergesort, line 4 Arguments: a = {4, 2, 1, 7, 6}, 1, 3 Result: Calling function: mergesort, line 4 Arguments: a = {4, 2, 1, 7, 6}, 1, 2 Result: Calling function: mergesort, line 5 Arguments: a = {4, 2, 1, 7, 6}, 2, 2 Result: {2}

17. Example (cont'd) Calling function: main, line 3 Arguments: a = {2, 4, 1, 7, 6}, 1, 6 Result: Calling function: mergesort, line 4 Arguments: a = {2, 4, 1, 7, 6}, 1, 3 Result: Calling Function: mergesort, line 5 Arguments: a = {2, 4, 1, 7, 6}, 3, 3 Result: {1}

18. Example (cont'd) Calling function: main, line 3 Arguments: a = {1, 2, 4, 7, 6}, 1, 6 Result: Calling function: mergesort, line 4 Arguments: a = {1, 2, 4, 7, 6}, 1, 3 Result: {1, 2, 4}

19. Example (cont'd) Calling function: main, line 3 Arguments: a = {1, 2, 4, 7, 6}, 1, 6 Result: Calling function: mergesort, line 5 Arguments: a = {1, 2, 4, 7, 6}, 4, 6 Result: