Recursion

1. Recursion

2. Canonical example: factorial • Recursion • When a method calls itself • Classical example – the factorial function • n! = 1·2·3·····(n-1)·n • Recursive definition

3. Example I using recursion: Sum of values from 1 to N • Problem • computing the sum of all the numbers between 1 and N • This problem can be recursively defined as:

4. Example I using recursion: Sum of values from 1 to N (ctd) // This method returns the sum of 1 to num // Refer to SumApp project public intsum (intnum) { int result; if (num == 1) result = 1; else result = num + sum (n-1); return result; }

5. result = 6 main sum(3) result = 3 sum sum(2) result = 1 sum sum(1) sum Example I using recursion: Sum of values from 1 to N (ctd)

6. Content of recursive definition • Base case (s) • Values of the input variables for which • no recursive calls are performed • There should be at least one base case • Every chain of recursive calls must • Eventually reach a base case • Recursive calls • Calls to the current method • Defined in such a way that • It makes progress towards a base case

7. Example II using recursion: Factorial • Factorial of a positive integer n, written n! is the product • n · (n – 1) · (n – 2) · … · 1 • with 1! equal to 1 and 0! defined to be 1 • The factorial of integer number can be calculated • iteratively (non-recursively) using a for statement as follows: • factorial = 1; • for(intcounter = number; counter >= 1; counter-- ) factorial *= counter; • Recursive declaration of the factorial method is arrived at • by observing the following relationship: • n! = n · (n – 1)!

8. Example II using recursion: Factorial (cont’d) • long should be used so that the program • can calculate factorials greater than 12! • Package java.math provides classes • BigIntegerand BigDecimalexplicitly for • high precision calculations not supported by primitive types. • Refer to FactorialApp project

9. Example II using recursion: Factorial (cont’d) • BigInteger method compareTo • compares the BigInteger that calls the method to • the method’s BigInteger argument. • Returns • -1 if the BigInteger that calls the method • is less than the argument, • 0 if they are equal or • 1 if the BigInteger that calls the method • is greater than the argument. • BigIntegerconstant • ONErepresents the integer value 1. • ZEROrepresents the integer value 0.

10. Example III using recursion: Fibonacci Series • The Fibonacci series, begins with 0 and 1 and • has the property that each subsequent Fibonacci number • is the sum of the previous two. • 0, 1, 1, 2, 3, 5, 8, 13, 21, … • The ratio of successive Fibonacci numbers converges to • a constant value of 1.618…, • called the golden ratio or the golden mean. • The Fibonacci series may be defined recursively: • fibonacci(0) = 0fibonacci(1) = 1fibonacci(n) = fibonacci(n–1) + fibonacci(n–2)

11. Example III using recursion: Fibonacci Series • Two base cases for Fibonacci method • fibonacci(0) is defined to be 0 • fibonacci(1) to be 1 • Fibonacci numbers tend to become large quickly. • We use type BigInteger as the thereturn type of • The fibonacci method • BigInteger methods multiply and subtract • implement multiplication and subtraction. • Refer to FibonacciApp project

12. return 4 * 6 = 24 final answer call recursiveFactorial ( 4 ) return 3 * 2 = 6 call recursiveFactorial ( 3 ) return 2 * 1 = 2 call recursiveFactorial ( 2 ) return 1 * 1 = 1 call recursiveFactorial ( 1 ) return 1 call recursiveFactorial ( 0 ) Visualizing Recursion for Factorial • Recursion trace • A box for each recursive call • An arrow from each caller • to callee • An arrow from each callee • to caller showing return value Example recursion trace:

13. Example IV using recursion: Binary Search • A binary search • Assumes the list of items in the search pool is sorted • Eliminates a large part of search pool with 1 comparison • Examines the middle element of the list • If it matches the target, the search is over • Otherwise, only one half of the remaining elements • Need to be searched • Then examines the middle element of remaining pool • Eventually, the target is found or data is exhausted

14. Example IV using recursion: Binary Search (cont’d) • Example: • Search a sorted array for a given value • BS(A, key, start, end) • Look for key in the array A • where elements are sorted according to ascending order • A method that calls itself with a smaller input set end: index of 1st element in search pool start: index of 1st element in search pool middle A

15. Binary search recursion: pseudo-code // Refer to BinarySearchApp project Boolean BS(A, key, start, end) mid = (start+end)/2 if(A[mid] == key) return true else if(end <= start) return false else if (A[mid] > key) return BS(A, key, start, mid-1) else return BS(A, key, mid+1, end)

16. Example V using recursion: Towers of Hanoi • Given • A platform with three pegs sticking out of it • Each peg is a stack that can accommodate n disks • Puzzle • Move all disks from peg a to peg c, one disk at a time • So that we never place a larger disk on top of smaller one • Refer to TowersOfHanoiApp project

17. Original Configuration Move 1 Move 2 Move 3 Towers of Hanoi

18. Move 4 Move 5 Move 6 Move 7 (done) Towers of Hanoi