Recursion. What is Recursion? Simple Example. Repeatedly Reduce the Problem Into Smaller Problems to Solve the Big Problem

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1 Recursion Repeatedly Reduce the Problem Into Smaller Problems to Solve the Big Problem What is Recursion? A problem is decomposed into smaller sub-problems, one or more of which are simpler versions of the original problem Implemented by having a method that activates itself (with different parameters) 2 Simple Example Given a non-negative integer number, write the digits of the number vertically Solution: If the number is less than 10, write it Else Write the digits of (number / 10) vertically Write the single digit (number % 10) 3

2 4 Implementation public static void writevertical (int number) if (number < 10) System.out.println (number); writevertical (number / 10); System.out.println (number % 10); Method Activation Records When a method is called, a block of memory called its activation record is reserved for it; containing: Values for all of its parameters Where to return when the method completes its execution When a method is called, its activation record is placed on a stack called the execution stack 5 Components of Recursive Methods A recursive call Activates the same method with simpler arguments One or more stopping cases Also known as base cases Stops the recursion and causes a return to the previous activation of the method and completion of the execution In the absence of a stopping case, infinite recursion will occur eventually resulting in a stack overflow error 6

3 7 Another example Computing the factorial of a positive integer: If the number is equal to 1, return the number Else, multiply the number by the factorial of 1 less than the current number and return the result Implementation public static int factorial (int number) if (number == 1) return number; return number * factorial (number 1); 8 Reasoning About Recursion We must check for 2 things: The recursion is not infinite The recursive function produces the correct result 9

4 10 Special Case: One Level Recursion Every case is either a stopping case or makes a recursive call to a stopping case, e.g.: public static double power (double x, int n) double product; int count; if (x == 0 && n <=0) throw new IllegalArgument Exception (. ); if (n >= 0) product = 1; for (count = 1; count <= n; count++) product := product * x; return product; return 1 / power (x, -n) General Case Here is a more general definition of power: public static double power (double x, int n) if (x == 0 && n <= 0) throw new IllegalArgumentException ( ); if (x==0) return 0; if (n==0) return 1; if (n > 0) return x * power (x, n-1); return 1 / power (x, -n); 11 Ensuring No Infinite Recursion Define a variant expression as a numeric expression based on a recursive parameter The value of the recursive expression must decrease on each recursive call by at least some amount Once the value is equal to (or smaller than) some small value, called the threshold, a stopping case is reached If we can define a variant expression with the above characteristics, then the recursive method is guaranteed to terminate 12

5 13 Correctness of Recursive Methods Proof based on Induction: First prove that there is no infinite recursion Prove the base step(s) by proving that the case(s) when no recursion is involved produce correct results Prove that the recursive cases produce a correct result if the recursive calls produce a correct result Recursion vs. Loops Recursion is generally more time consuming because of the overhead of function calls Very deep recursion may cause a stack overflow and should be avoided. This requires that we estimate the depth of recursion of any recursive method Tail recursion (recursion towards the end of the method) can be easily avoided and should be avoided Some problems naturally lend themselves to recursion Some programming languages (such as Lisp) rely heavily on recursion and their compilers automatically convert recursive calls into loops in most cases 14

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