1. CSC 221: Recursion

2. Recursion: Definition Function that solves a problem by relying on itself to compute the correct solution for a smaller version of the problem Requires terminating condition: Case for which recursion is no longer needed

3. Recursion: Induction Basis Mathematical Induction: –Prove that statement is true for first n values, given that it is true for first n-1 values –Prove that the statement is true for a base case.

4. Recursion: Mathematical Induction Sum of first N positive integers is (N*(N+1)) / 2 Base Case: –1 st positive integer 1 (1 * (1+1)) / 2=>(1*2)/2=> 2/2 => 1 Inductive Case: Assume true for n-1 –Sum(1..N-1) = ((N-1) * (N-1+1)) / 2 => ((N-1) * (N)) / 2) = (N 2 –N)/2 –Adding N = (N 2 –N)/2 + N = (N 2 -N)/2 + 2N/2 = (N 2 + N)/2=> (N * (N+1)) / 2

5. Factorial Recursion Factorial:n! = n * (n-1)! –Base Case => 0! = 1 –Smaller problem => Solving (n-1)! Implementation: long factorial(long inputValue) { if (inputValue == 0) return 1; else return inputValue * factorial(inputValue - 1); }

6. Searching We want to find whether or not an input value is in a sorted list: 8 in [1, 2, 8, 10, 15, 32, 63, 64]? 33 in [1, 2, 8, 10, 15, 32, 63, 64]?

7. Searching int index = 0; while (index < listSize) { if (list[index] == input) return index; index++; } return –1;

8. Searching Better method: –Number of operations to find input if in the list: Dependent on position in list 1 operation to size of list –Number of operations to find input if not in the list: Size of list

9. Searching Better method? Use fact that we know the list is sorted Cut what we have to search in half each time –Compare input to middle –If input greater than middle, our value has be in the elements on the right side of the middle element –If input less than middle, our value has to be in the elements on the left side of the middle element –If input equals middle, we found the element.

10. Searching: Binary Search for (int left = 0, right = n –1; left <= right;) { middle =(left + right) / 2; if (input == list[middle]) return middle; else if (input < list[middle]) right = middle – 1; else left = middle + 1; } return – 1;

11. Searching: Binary Search 8 in [1, 2, 8, 10, 15, 32, 63, 64]? 1 st iteration: Left = 0, Right = 7, Middle = 3, List[Middle] = 10 Check 8 == 10 => No, 8 < 10 2 nd iteration: Left = 0, Right = 2, Middle = 1, List[Middle] = 2 Check 8 == 2 => No, 8 > 2 3 rd iteration: Left = 2, Right = 2, Middle = 2 Check 8 == 8 => Yes, Found It!

12. Searching: Binary Search Binary Search Method: –Number of operations to find input if in the list: Dependent on position in list 1 operation if middle Log 2 n operations maximum –Number of operations to find input if not in the list: Log 2 n operations maximum

13. Recursive Binary Search Two requirements for recursion: –Same algorithm, smaller problem –Termination condition Binary search? –Search in half of previous array –Stop when down to one element

14. Recursive Binary Search int BinarySearch(int *list, const int input, const int left, const int right) { if (left < right) { middle =(left + right) / 2; if (input == list[middle]) return middle; else if (input < list[middle]) return BinarySearch(list, input, left, middle-1); else return BinarySearch(list,input,middle+1,right); } return – 1; }

15. While vs Recursion While and Recursion are essentially interchangeable Considerations: –Efficiency –Simplification of programming –Readability/Understandability While they are equivalent, there is not always an obvious while implementation of some functions that are easily implemented with recursion

16. Fibonacci Computation Fibonacci Sequence: –1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … –Simple definition: –Fib[0] = 1 –Fib[1] = 1 –Fib[N] = Fib(N-1) + Fib(N-2)

17. Recursive Fibonacci int fibonacci(int input) { if ((input == 0) || (input == 1)) return 1; else return (fibonacci(input-1) + fibonacci(input-2)); }

18. Iterative Fibonacci int fibonacci(int input) { int first = 1; int second = 1; int temp; for (int k = 0; k < input; k++) { temp = first; first = second; second = temp + second; } return first; }

19. Efficiency of Recursion Recursion can sometimes be slower than iterative code Two main reasons: –Program stack usage –Result generation

20. Types of Recursion Linear Recursion: –1 recursive call per function –Factorial, Binary Search examples Tree Recursion: –2 or more recursive calls per function –Fibonacci Example

21. Efficiency of Recursion Stack Usage: –When a function is called by a program, that function is placed on the program call stack: main() getData() readFile() Returns file data to be used in getData() Returns formatted data to be printed in main()

22. Efficiency of Recursion Every stack entry maintains information about the function: –Where to return to when the function completes –Storage for local variables –Pointers or copies of arguments passed in

23. Efficiency of Recursion When using recursive functions, every recursive call is added to the stack and it grows fast: Fibonacci (5) –Fibonacci (5) = Fibonacci (4) + Fibonacci(3) –Fibonacci (1) in the computation of Fibonacci(4) is the first time we don’t have to call the function again. Stack entries take up space and require extra processing main() Fibonacci(5) Fibonacci(4) Fibonacci(3) Fibonacci(2) Fibonacci(1)

24. Efficiency of Recursion Another Reason for Slowdowns [Tree Recursion] –Traditional Recursion doesn’t save answers as it executes –Fib(5) = Fib(4) + Fib(3) = Fib(3) + Fib(2) + Fib(3) = Fib(2) + Fib(1) + Fib(2) + Fib(3) = Fib(1) + Fib(0) + Fib(1) + Fib(2) + Fib(3) = Fib(1) + Fib(0) + Fib(1) + Fib(1) + Fib(0) + Fib(3) = Fib(1) + Fib(0) + Fib(1) + Fib(1) + Fib(0) + Fib(2) + Fib(1) = Fib(1) + Fib(0) + Fib(1) + Fib(1) + Fib(0) + Fib(1) + Fib(0) + Fib(1) Solution: Dynamic programming – saving answers as you go and reusing them

25. Dynamic Programming Fibonacci Problem –We know the upper bound we are solving for –Ie Fibonacci (60) = 60 different answers –Generate an array 60 long and initalize to –1 –Everytime we find a solution, fill it in in the array –Next time we look for a solution, if the value in the array for the factorial we need is not –1, use the value present.

26. Fibonacci Examples Three implementations of fibonacci: –Naive recursion implementation (worst performance) –Dynamic Programming recursion implementation (better) –Iterative implementation (best)

27. Recursive Datastructures Recursion is useful when datastructure is inherently recursive Unix directory hierarchy is a tree datastructure / /home /var/usr /home/turketwh /home/turketwh/CS112/home/turketwh/CS221

28. Directory Traversal ls –R in Unix=> recursive list Potential implementation? –listDirectory(directory baseDirectory) { file[] files = getFiles(); int fileCount = getFileCount(); for (int i = 0; i < fileCount; i++) { if (files[I].type == “dir”) listDirectory(file); else listFile(file); }

29. Recursive Datastructures Will see a lot of datastructures that are recursive –Lists [Atom + Smaller List] –Trees [ Root + Subtrees] Have simple implementations because their functionality can be defined recursively.

30. Towers of Hanoi Not allowed General Problem: For any number of boxes, move boxes from start peg to destination peg. Can never place a bigger box on top of a smaller box.

31. Towers Of Hanoi Recursive? 1 box from peg 1 to peg 3 1 _ __ _ 1 2 boxes from peg 1 to peg 3 1 2 __2 1 __ 1 2_ _ 2

32. Towers Of Hanoi 3 boxes from peg 1 to peg 3 1 2 2 1 1 3 _ _ 3 _ 13 2 13 2 __ 2 3 1 2 1 2 31 _ 3_ _ 3 For n boxes, 1)Solve the n-1 problem from start to the temp peg 2)Move the nth box to the destination peg 3)Solve the n-1 problem from the temp peg to the destination peg

33. Space Efficiency Factorial(N) –Factorial(4) => =4*Factorial(3) =4 * 3 * Factorial(2) = 4 * 3 * 2 * Factorial(1) =4 * 3 * 2 * 1 –Has to make a maximum of N calls before base case is reached and function returns. –N activation records will be placed on the stack. Since the size of the input is N, this function requires memory that is linearly related to the size of the input

34. Space Efficiency Fibonacci(N) – Standard recursive implementation –Fib(5) = Fib(4) + Fib(3) –For Fib(4), also puts Fib(3),Fib(2), Fib(1) on stack –For Fib(3), also puts Fib(2), Fib(1) on stack –Fib(4), Fib(3) compute separately Space Efficiency considers maximum amount required at one time – Fib(4) branch in this case Requires memory linearly related to input size

35. Tail Recursion Tail Recursion: When the results of a recursive call are not used after the recursive call within the calling function. long factorialHelper(long startValue, long inputValue) { if (startValue == 0) return 1; else if (inputValue == 1) return startValue; else return factorialHelper(startValue * (inputValue-1), inputValue - 1); } long factorial(long inputValue) { return factorialHelp(inputValue, inputValue); }

36. Space Efficiency: Tail Recursion Since the results of the recursive call aren’t needed for other computations, the stack frames aren’t needed to hold partial results. With each call, new frame replaces old frame. –The compiler handles these optimizations. Requires constant amount of memory and is not dependent on the size of the input.

37. Space Efficiency: Tail Recursion main() factorialHelper(3,3) factorialHelper(6,2) factorialHelper(6,1) Without Optimization Return 6 Call With Optimization looks like this: (each call of fh is done when it calls the next, so take it off the stack) main() factorialHelper(3,3) main() factorialHelper(6,2) main() factorialHelper(6,1)

38. Recursion Key Ideas –Decomposition: Solve smaller problem(s) and combine answers –Tends to allow for very simple writing of code –Used naively, may lead to Significant stack usage Repeated computations –Have touched on methodologies to help work around those issues: Tail Recursion Storing answers (‘dynamic programming’) –Suggests a new technique may be needed for computing number of operations for algorithm to complete