1. 1 Recursion Dr. Bernard Chen Ph.D. University of Central Arkansas

2. 2 Mathemetical Induction To prove Let p(n) denote the statement involving the integer variable n. The Principle of Mathematical Induction states: If p(1) is true and, for some integer K >=1, p(k+1) is true whenever p(k) is true then p(n) is true for all n>=1. 4 steps in using Induction: Base cases; --- p(1), p(2), … Induction hypothesis (IH); --- assume p(k) is true Statement to be proved in induction; --- it is true for p(k+1) Induction step.--- prove p(k+1) is true based on IH

3. 3 Recursive Function Call a recursion function is a function that either directly or indirectly makes a call to itself. but we need to avoid making an infinite sequence of function calls (infinite recursion)

4. 4 Finding a Recursive Solution a recursive solution to a problem must be written carefully the idea is for each successive recursive call to bring you one step closer to a situation in which the problem can easily be solved this easily solved situation is called the base case each recursive algorithm must have at least one base case, as well as a general (recursive) case

5. 5 General format for Many Recursive Functions if (some easily-solved condition) // base case solution statement else // general case recursive function call

6. 6 A recursive definition int s (int n) { if (n ==1) return 1; else return s(n-1) + n; } A few of problems: n  0 at the beginning; return value might be too large to fit in an int.

7. 7 When a function is called... a transfer of control occurs from the calling block to the code of the function--it is necessary that there be a return to the correct place in the calling block after the function code is executed; this correct place is called the return address when any function is called, the run-time stack is used--on this stack is placed an activation record for the function call

8. 8 Stack Activation Frames the activation record contains the return address for this function call, and also the parameters, and local variables, and space for the function’s return value, if non-void the activation record for a particular function call is popped off the run-time stack when the final closing brace in the function code is reached, or when a return statement is reached in the function code at this time the function’s return value, if non-void, is brought back to the calling block return address for use there

9. 9 A Stake of Activation Records S(1) S(2) S(3) S(4) Main()

10. 10 int Func ( /* in */ int a, /* in */ int b ) { int result; if ( b == 0 ) // base case result = 0; else if ( b > 0 ) // first general case result = a + Func ( a, b - 1 ) ) ; // instruction 50 return result; } A recursive function

11. 11 FCTVAL ? result ? b 2 a 5 Return Address 100 Run-Time Stack Activation Records x = Func(5, 2);// original call at instruction 100 original call at instruction 100 pushes on this record for Func(5,2)

12. 12 FCTVAL ? result ? b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 record for Func(5,2) call in Func(5,2) code at instruction 50 pushes on this record for Func(5,1) Run-Time Stack Activation Records x = Func(5, 2);// original call at instruction 100

13. 13 FCTVAL ? result ? b 0 a 5 Return Address 50 FCTVAL ? result 5+Func(5,0) = ? b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 record for Func(5,2) record for Func(5,1) call in Func(5,1) code at instruction 50 pushes on this record for Func(5,0) Run-Time Stack Activation Records x = Func(5, 2);// original call at instruction 100

14. 14 FCTVAL 0 result 0 b 0 a 5 Return Address 50 FCTVAL ? result 5+Func(5,0) = ? b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 record for Func(5,0) is popped first with its FCTVAL record for Func(5,2) record for Func(5,1) Run-Time Stack Activation Records x = Func(5, 2);// original call at instruction 100

15. 15 record for Func(5,2) record for Func(5,1) is popped next with its FCTVAL FCTVAL 5 result 5+Func(5,0) = 5+ 0 b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 Run-Time Stack Activation Records x = Func(5, 2);// original call at instruction 100

16. 16 FCTVAL 10 result 5+Func(5,1) = 5+5 b 2 a 5 Return Address 100 record for Func(5,2) is popped last with its FCTVAL Run-Time Stack Activation Records x = Func(5, 2);// original call at instruction 100

17. Let’s try an example 17

18. 18 Too much recursion Can Be Dangerous Fibonacci numbers. Long fib (int n) { If (n <=1) return n; Else return fib(n-1) + fib(n-2); }

19. 19 Too much Recursion Can be Dangerous This definition will lead to exponential running time. Reason: -- too much redundant work Not necessary to use recursive Write a non-recursive Fibonacci code

20. 20 Fibonacci

21. 21 Tree Tree is a fundamental structure in computer science. Recursive definition: A tree is a root and zero or more nonempty subtrees. Nonrecursive definition: A tree consists of s set of nodes and a set of directed edges that connect pairs of nodes. That is, a connected graph without loop.

22. 22 Traversal Three standard traversal order preorder - V L R inorder - L V R postorder - L R V Preorder: traverse the node itself first, then all nodes in the LEFT subtree, then all nodes in the RIGHT subtree. Inorder: traverse all nodes in the LEFT subtree first, then the node itself, then all nodes in the RIGHT subtree. Postorder: traverse all nodes in the LEFT subtree first, then all nodes in the RIGHT subtree, then the node itself, V RL

23. 23 Recursive Traversal Implementation Void PrintInorder (root) if root != null PrintInorder(root->left); print(root->data); PrintInorder(root->right); endif; Void PrintInorder (root) if root != null PrintInorder(root->left); print(root->data); PrintInorder(root->right); endif; The difference is the order of the three statements in the ‘IF’. 1 23 456 preorder : 1 2 4 5 3 6 inorder : 4 2 5 1 3 6 postorder : 4 5 2 6 3 1 preorder : 1 2 4 5 3 6 inorder : 4 2 5 1 3 6 postorder : 4 5 2 6 3 1 Void PrintPreorder (root) if root != null print(root->data); PrintPreorder(root->left); PrintPreorder(root->right); endif; Void PrintPreorder (root) if root != null print(root->data); PrintPreorder(root->left); PrintPreorder(root->right); endif; Void PrintPostorder (root) if root != null PrintPostorder(root->left); PrintPostorder(root->right); print(root->data); endif; Void PrintPostorder (root) if root != null PrintPostorder(root->left); PrintPostorder(root->right); print(root->data); endif;

24. Binary Search 3-ways comparisons int binarySearch(vector a[], int x){ int low = 0; int high = a.size() – 1; int mid; while(low <= high) { mid = (low + high) / 2; if(a[mid] < x) low = mid + 1; else if( a[mid] > x) high = mid - 1; else return mid; } return NOT_FOUND; // NOT_FOUND = -1 }//binary search using three-ways comparisons

25. 25 Some more examples: Factorials Binary Search template int binarySearch( const vector & a, const Comparable & x, int low, int high ) { if( low > high ) return NOT_FOUND; int mid = ( low + high ) / 2; if( a[ mid ] < x ) return binarySearch( a, x, mid + 1, high ); else if( x < a[ mid ] ) return binarySearch( a, x, low, mid - 1 ); else return mid; }

26. 26 Divide and Conquer Given an instance of the problem to be solved, split this into several, smaller, sub-instances (of the same problem) independently solve each of the sub-instances and then combine the sub- instance solutions so as to yield a solution for the original instance.

27. 27 Recursion or Iteration? EFFICIENCYCLARITY