Recursion

1. Recursion Chapter 11

2. How it works • “A recursive computation solves a problem by using the solution of the same problem with simpler inputs” Big Java, Horstman, p. 669 • “For recursion to terminate, there must be special cases for the simplest inputs.” Big Java, Horstman, p. 670

3. An example public class Triangle { private int aWidth; public Triangle (int width) { aWidth = width; } public int getArea() { } //Big Java, Horstman, p. 670-71

4. An example public class Triangle { private int aWidth; public Triangle (int width) { aWidth = width; } public int getArea() { if (width <= 0) return 0; if (width ==1) return 1; Triangle smallerTriangle = new Triangle(width -1); int smaller Area = smallerTriangle.getArea(); return smallerArea + width; }

5. Common Error • Infinite recursion!!!!

6. Key requirements for Recursion • Every recursive call must simplify the computation in some way • There must be special case(s) to handle the simplest computations directly.

7. Alternative to recursion • Iteration • Why one or the other? • “In many cases a recursive solution is easier to understand and implement correctly than an iterative solution. • Occasionally, a recursive solution runs much slower than its iterative counterpart. However, in most cases, the recursive solution is only slightly slower.” Big Java, Horstman, p. 694

8. Recursive Definitions • “ A definition in which the item being defined appears as part of the definition is called an inductive definition or a recursive definition… • Recursive definitions have two parts: • A basis, where some simple cases of the item being defined are explicitly given • An inductive or recursive step, where new cases of the item being defined are given in terms of previous cases” Mathematical Structures for Computer Science, Gersting, pp.120-121

9. Example of a Sequence • A sequence S is a list of objects that are enumerated (listed) in some order; there is a first such object, then a second, and so on. S(k) denotes the kth object in the sequence. A sequence is defined recursively by explicitly naming the first value (or the first few values) in the sequence and then defining later values in the sequence in terms of earlier values.

10. Explicit Examples – recursively defined sequences • Example 1 S(1) = 2 S(n) = 2S(n-1) for n>= 2 • Example 2 T(1) = 1 T(n) = T(n-1) + 3 for n >= 2 • Example 3 F(1) = 1 F(2) = 1 F(n) = F(n-1) + F(n-2) for n > 2

11. Explicit Examplesrecursively defined operations • Example 4 a0 = 1 an = (an-1)a for n >= 1 • Example 5 m(1) = m m(n) = m(n-1) + m for n >= 2

12. Code Example • Example 6 public int factorial(int n) { int value; if ((n == 0) || (n == 1)) value = 1; else value = n*factorial(n-1); return value; }

13. Computer Behavior • When a method is called, an activation record (or invocation record) is created • Method calls/returns happen in a last-in-first-out manner • Activation Records Contain: • Local variables and their values • The location (in the caller) of the function call • Execution status (e.g., processor registers) of the current module • Other things

14. Card technique for understanding recursion • Use "Cards" for Activation Records: • Each card contains the local variables and their values • The location (in the caller) of the method call • Tracing Execution: • Add a "card" to the top of a stack when a method is called and remove a "card" when the method returns

15. Designing Recursive Algorithms • An Example - Print the Digits of a Number: • Easy Case - The number consists of 1 digit • Reducing Difficult Cases - Integer division by 10

16. Implementation – Print the Digits of a Number: • public void printBase10(int n) { if (n >= 10) printBase10(n/10); System.out.print(n%10); }