Recursion

2. Recursion

3. Recursion You should be able to identify the base case(s) and the general case in a recursive definition • To be able to write a recursive algorithm for a problem involving only simple variables.

4. A Recursive call • A function call in which the function being called is the same as the one making the call. • Put differently it is a function call to itself. • When a function calls itself this is said to be a recursive call. • The word recursive means: having the characteristics of coming up again or repeating. • Can be used instead of iteration or looping

5. Efficiency considerations • Recursive solutions are generally less efficient than iterative solutions. • Some problems that we will see lend themselves to recursion • Some languages do not allow recursion e.g. Fortran, Basic and Cobol • Lisp on the other hand is especially suited for recursive algorithms • Initially we will look at recursive algorithms on simple variables we will then look at recursion on structured data

6. What is recursion? • The simplest use of the recursive algorithm is to compute a number raised to a given power, e.g. 2^3 is 2x2x2, 4^5 is 4x4x4x4x4 and so on. • y^n, yxyxyxyxyxyx…. (n times) • y^n is yxy^(n-1) • y^n is yxyxy^(n-2) • etc

7. The recursive definition • This is the important bit. • We can write x^n as x * x^(n-1). Hence something is defined in terms of a smaller version of itself • Base case: the case for which the solution can be stated non recursively • General case: the case for which the solution is expressed in terms of a smaller version of itself

8. See program power.cpp demo • Run power.cpp • See also power*.cpp in the cpteach directory included in the week 8 directory • See all Visual Basic illustrations in this directory go t examples directory

9. Some comments on the program • Each recursive call to Power can be thought of as creating its own copy of the parameters x and n. • x remains the same for each call but the value of n decreases each time. • Lets consider the case with number =2 and exponent = 3, so we are computing 2^3 which is 8

10. Tracing through the Power program • Call 1. Power is called by main, with number equal to 2 and exponent equal to 3. • Because n does not equal 1 Power is called recursively with x and n-1as arguments. • Call 2: x is equal to 2 and n is equal to 2. Because n does not equal 1 the function Power is called recursively. • Call 3: x = 2 and n = 1. As n=1, the value of x is returned. This call to the function has terminated

11. Infinite recursion • If the function Power was called with n negative then the condition n==1 would never be satisfied since starting at a value less than zero and continually subtracting 1 would mean that a value of 1 is never obtained. • I have altered this now so that negative powers are calculated

12. Execution of Power(2,3) Power(2,3) Call 1 X=2, n=3 X=2, n=2 Call 2 X=2, n=1 Call 3

13. Recursive algorithms with Simple Variables • Lets look at a special mathematical function important in so many areas of mathematics. It is the factorial function defined as • n! = nx(n-1)x(n-2)x……1 • E.g. 4! = 4*3*2*1 = 24

14. Run Factorial demoFactorial.cpp

15. Tracing through the Factorial function Call 1 n=4 Call 2 n=3 n=2 Call 3 n=1 Call 4 n=0 Call 5

16. Iterative solution Writing the code without using recursion gives: int Factorial(int n) { int factor; int count; Factor = 1; for (count=2;count<=n;count++) Factor=Factor*count; return Factor; }

17. The Towers of Hanoi • Very old problem demonstrating recursion. • Only one disc can be moved at a time • After each move, all discs must be on the poles • No disc may be placed on top of a disc which is smaller than itself

18. Hanoi.cpp • Discuss Hanoi.cpp and demonstrate