1. Thursday, January 25, 2007 "To define recursion, we must first define recursion." - Anonymous

2. Remaining Lectures Schedule §Thursday January 25 (Today) §Friday January 26 §Thursday February 1 §Friday February 2 §Saturday February 3 §Tuesday February 6

3. What is wrong here? int* function_returns_ptr(void){ int array[8]={1,2,3,4,5,6,7,8}; int *ptrArray=array; return (ptrArray+4); } int main(void){ int *result=function_returns_ptr(); cout<<*result<<endl; return 0; }

4. References Safer and simpler use of memory addresses are references When a reference parameter is used, the address of an argument is automatically passed to the function. Within the function the operations on reference parameter are automatically de-referenced.

5. §With call by reference, the caller gives the called function the ability to directly access the caller’s data and to modify that data if the called function so chooses. §To indicate that a function parameter is passed by reference, simply follow the parameter’s type in the function prototype by an ampersand &.  e.g. void myFunction( int &x); Calling Functions

6. int squareByValue(int a); void squareByReference(int &aref); Calling Functions

7. int main() { int x = 2, z = 4; cout << "x = " << x << " before squareByValue" ; squareByValue(x) ; cout<< "x = " << x << " after squareByValue" ; cout << "z = " << z << "before squareByReference”; squareByReference(z); cout << "z = " << z << " after squareByReference"; return 0; } Calling Functions

8. int squareByValue(int a) { return a *= a; //caller's argument not modified } void squareByReference(int &cRef) { cRef *= cRef; // caller's argument modified } Calling Functions

9. §Function to swap two numbers

10. void swap(int &sa, int &sb){ int temp=sa; sa=sb; sb=temp; } int main() { int a=3; int b=5; cout<<a<<" "<<b<<endl; swap(a, b); cout<<a<<" "<<b<<endl; return 0; } Calling Functions -II

11. void f1(int *num) { cout<<*num<<endl; num++; cout<<*num<<endl; } int main(void){ int a[8]={1,2,3,4,5,6,7,8}; int *aptr=a; f1(aptr); cout<<*aptr<<endl; return 0;} //OUTPUT? Calling Functions with pointers

12. 131131

13. void f1(int* &num) { cout<<*num<<endl; num++; cout<<*num<<endl; } int main(void){ int a[8]={1,2,3,4,5,6,7,8}; int *aptr=a; f1(aptr); cout<<*aptr<<endl; return 0;} //OUTPUT?

14. 133133

15. The most common use of references is in function parameters.

16. int a=6; int &b =a; cout<<b<<" "<<a<<endl; b=3; cout<<b<<" "<<a<<endl; References must be initialized Reference variables

17. 6 3 Reference variables

18. What is wrong here? int a=6; int &b=a+4; cout<<b<<" "<<a<<endl; Reference variables

19. A function call ends when l Return statement is executed. l The closing } brace of the end of function is reached. §At the end of function call, flow of control returns back to the calling function.

20. void X() { int a = 1; int b = 2; // T1 Y(a); // T3 Y(b); // T5 } void Y(int p) { int q; q = p + 2; // T2 (first time through), T4 (second time through) }

21. §Remember the even odd question?

22. Recursion A recursive function is one that calls itself. Recursive functions have two parts §Stopping Condition(s) -- solving the base case(s). §Recursive call -- breaking the problem down.

23. Recursion When a function calls itself, new local variables and parameters are allocated storage on stack and the function is executed with these new variables from the beginning of function.

24. void PrintFunc(int x) {int y=x-1; if (x<=0) return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl; } int main(void) { PrintFunc(3); //... }

25. §Focus on the flow of control

26. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3

27. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3 x is 2

28. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3 x is 2 x is 1

29. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3 x is 2 x is 1 x is 0

30. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3 x is 2 x is 1

31. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3 x is 2

32. void PrintFunc(int x) {int y=x-1; if (x<=0)return; cout<<"In Print Func before recursive call. x="<<x<<endl; PrintFunc(y); cout<<"In Print Func after recursive call. x="<<x<<endl;} x is 3

33. In Print Func before recursive call. x=3 In Print Func before recursive call. x=2 In Print Func before recursive call. x=1 In Print Func after recursive call. x=1 In Print Func after recursive call. x=2 In Print Func after recursive call. x=3

34. int RecFunc(int x){ int sum; if(x<=1) return 1; sum = x*x + RecFunc(x-1); cout<<sum<<endl; return sum; } int main(void){ int ans; ans=RecFunc(5); cout<<"ans="<<ans<<endl; //... }

35. 5 14 30 55 ans=55

36. Recursion vs. Iteration Similarities: Both iteration and recursion involve repetition: §Iteration explicitly uses a repetition structure; §recursion achieves repetition through repeated function calls. Iteration and recursion each involve a termination test §iteration terminates when the loop condition fails § recursion terminates when a base case is recognized.

37. Recursion vs. Iteration Similarities: Both iteration and recursion can occur infinitely: §An infinite loop can occur with iteration, §Infinite recursion can occur when the base case is not reached.

38. SELF TEST: Recursion int factorial(int N){ int ans; if (N<=1) return 1; //base case ans=factorial(N-1)*N; return ans; } int main() { int answer=factorial(4); cout<<answer; return 0; } Stack contents

39. Example: Binary Search int BinarySearch(int no, int low,int high, int* iarray); int main(void) { int num, i, size=15; int* array=new int[size]; for (i=0; i<size; i++) array[i]=13*i+4; for (i=0; i<size; i++) cout<<array[i]<<" "; cout<<endl; cin>>num; if(BinarySearch(num, 0, size-1, array)) cout<<"found"<<endl; else cout<<"not found"<<endl; return 0; }

40. Example: Binary Search int BinarySearch(int no, int low,int high, int* iarray){ if (low>high){ return 0; } int mid=low+(high-low)/2; if (no==iarray[mid]) return 1; else if (no>iarray[mid]) return BinarySearch(no, mid+1, high, iarray); else return BinarySearch(no, low, mid-1, iarray); }