1. Recursion Definition: A method that calls itself. Recursion can be direct or indirect. Indirect recursion involves a method call that eventually recalls the original method Examples public factorial(int n) { if (n==1) return 1; return n * factorial(n-1); } public hello(int n) { if (n>=0) { System.out.prinln("Hello"); hello(n-1); }

2. Why does it work? Java creates an activation record for every method call Definition: An activation record is a block of memory on the Java stack containing parameter values, local variables, the return address, and return value. The activation only exists while the method executes Java releases all local storage when the method returns A stack is a last-in first-out list A list is an ordered collection of elements. Example: do(9, -1); Method public int do(int n,int k) { String str = "abc"; int num = 5; return str.length()+num; } Activation Record n: 9 k: -1 str: "abc" value: 5 return: instruction after the call

3. Another Example Simple Example: Sum(int size, int[] x) { if size>0) return x[size]+sum(size-1); return x[0]; } Notes: –The 'if' statement is the non-recursive part (the base case) –The 'second return' is the recursive step –Infinite loop occurs if the base case never executes Question: –What happens if size is negative?

4. Designing a recursive method Three are two requirements –Define the base case –Define the relationship between the problem and into one or more smaller problems of the same kind When is recursion beneficial? –Recursive algorithms are ALWAYS slower than their non- recursive counterparts. Why? –Recursive algorithms can often reduce a problem to just a few statements. –Non-recursive algorithms can be sometimes more difficult to program and maintain. –Recursion is useful when the recursion step significantly reduces the problem size. Normally, static variables should not be modified by a recursive method. Why?

5. Some more simple examples Which are good uses of recursion? Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, … –Base case: n<=2; Recursive step: f(n) <- f(n-2)+f(n-1) Greatest common denominator –Base case: y%x = 0; Recursive step: gcd(x,y) <- gcd(y%x,x) Palindrome Tester –Base case: length last character –Recursive step: Pal(s.substring(1,s.length()-1))<-Pals(s) Print a string (Question: How would you reverse a string?) –Base case: length=1; Recursive step: print char 0 and call with remainder Count string characters, translate a string, sequential search, print all head and tail outcomes Tower of Hanoi (How many moves to solve the 64 problem?) –Base case: Number of pegs <=1 –Recursive step: H(n) <- Move n-1 from source to spare, move last peg to destination, Move n-1 from spare to destination

6. Traversing a maze Base case –Current location in the array is [n-1,n-1] Recursive step –For each direction traverse maze in that direction. Return result if true returned as result of the recursive call Enhancement possibilities –Transform the problem to three dimensions? –How can the original maze be preserved? –How could we print the solution path? –How could we find the shortest path solution?

7. Graphics and Fractals Definition: A recursive geometric pattern Examples: –Drawing a tree visually –Sierpinski's triangles –Koch's Snowflake –Mandelbrot set Our lab4: Generate a recursive landscape