Recursion

1. Recursion Chapter 11

2. Chapter Contents • Recursive Definitions • Base and General Cases of Recursion • What is a Recursive Algorithm • Recursive Functions • Using Recursive Functions

3. Recursive Definitions • Definition:The process of solving a problem by reducing it to smaller versions of itself is called recursion. • Example:Consider the concept of factorials 0! = 1 1! = 1 2! = 1 * 2 3! = 1 * 2 * 3 . . . n! = 1 * 2 * 3 * 4 * … * (n – 1) * n

4. Base Case General Case Recursive Algorithm • The factorial of a number could be defined recursively

5. Base Case General Case Base and General Cases of Recursion • Every recursive definition must have one (or more) base cases. • The general case must eventually reduce to a base case. • The base case stops the recursion

6. Recursive Functions • This would be the recursive function for the recursive factorial algorithm int fact (int num) { if (num == 0) return 1; else return num * fact (num – 1);}

7. Recursive Functions • Execution ofa call to therecursive factorial functioncout << fact(4);

8. Recursive Functions • Think of a recursive function as having infinitely many copies of itself. • Every call to a recursive function has • its own code and • its own set of parameters and • its own local variables. • After completing a particular recursive call, • Control goes back to the calling environment • That is, the previous call.

9. Recursive Functions • The current (recursive) call must execute completely before the control goes back to the previous call. • The execution in the previous call begins from the point immediately following the recursive call. • A recursive function in which the last statement executed is the recursive call is called a tail recursive function. • The function fact is an example of a tail recursive function.

10. Using Recursive Functions • Consider the task of finding the largest element of an array • To find the largest element in list[a]...list[b] • First find the largest element in list[a+1]...list[b] • Then compare this largest element with list[a]. • Note the source code

11. Using Recursive Functions • Execution ofthe recursivefunctionlargest()

12. Using Recursive Functions • Example of Fibonacci number • a denotes the first Fibonacci number • b is the second Fibonacci number • n is the nth Fibonacci number:

13. Using Recursive Functions • Note Fibonacci source code • Note diagram ofrecursiveFibonacci function call • Very inefficient

14. Recursion or Iteration? • Iterative control structures use a looping structure to repeat a set of statements. • while, • for, • or do...while, • There are usually two ways to solve a particular problem • iteration and • recursion. • The obvious question … Which method is better?

15. Recursion or Iteration? • Another key factor in determining the best solution method is efficiency. • Recall that whenever a function is called, • Memory space for its formal parameters and local variables is allocated. • When the function terminates, that memory space is then deallocated. • We also know that every (recursive) call has its own set of parameters and (automatic) local variables.

16. Recursion or Iteration? • Overhead involved with recursion • Memory space • Time of execution • Today’s computers • fast • inexpensive memory • Therefore, the execution of a recursion function is not noticeable.

17. Recursion or Iteration? • Rule of thumb:If an iterative solution is • more obvious • easier to understand than a recursive solution Use the iterative solution, which would be more efficient. • When the recursive solution is more obvious or easier to construct … use recursion Fibonacci numbers Tower of Hanoi

18. A Recursive Polynomial • The function below is an approximation for ex • Can be defined recursively