Recursion

1. Recursion Gordon College CPS212 Adapted from Nyhoff: ADTs, Data Structures, and Problem Solving

2. Recursive Functions • Recursive programming is based off of Recursive Formulas • Definition of Recursive Formula • a formula that is used to determine the next term of a sequence using one or more of the preceding terms. • Example of Recursive Formula • The recursive formula for the sequence 5, 20, 80, 320, ... is an = 4 an-1 How would you program such a thing?

3. Program Solution int series(int n) { if (n == 1) return 5; return 4 * series(n-1); } Base case Recursive case

4. Recursion A function is defined recursively if it has the following two parts: • An anchor or base case • The function is defined for one or more specific values of the parameter(s) • An inductive or recursive case • The function's value for current parameter(s) is defined in terms of previously defined function values and/or parameter(s)

5. Recursive Example • Consider a recursive power functiondouble power (double x, unsigned n){ if ( n == 0 ) return 1.0; else return x * power (x, n-1); } • Which is the anchor? • Which is the inductive or recursive part? • How does the anchor keep it from going forever?

6. Recursive Example • Note the results of a call • Recursivecalls • Resolutionof thecalls

7. Recursive Factorialanother example • n! = 1 x 2 x …x n, for n > 0 n! = (n – 1)! X n 5! = 5 x 4! 120 4! = 4 x 3! 24 3! = 3 x 2! 6 2! = 2 x 1! 2 1! = 1 1

8. Another Example of Recursion • Fibonacci numbers1, 1, 2, 3, 5, 8, 13, 21, 34f1 = 1, f2 = 1 … fn = fn -2 + fn -1 • A recursive functiondouble Fib (unsigned n){ if (n <= 2) return 1; else return Fib (n – 1) + Fib (n – 2); }

9. A Bad Use of Recursion • Why is this inefficient? • Note the recursion tree

10. Uses of Recursion • Binary Search • See source code • Note results of recursive call

11. Uses of Recursion • Palindrome checker • A palindrome has same value with characters reversed1234321 racecar Recursive algorithm for an integer palindrome checker If numDigits <= 1 return true Else check first and last digits num/10numDigits-1and num % 10 • if they do not match return false If they match, check more digitsApply algorithm recursively to:(num % 10numDigits-1)/10 and numDigits - 2

12. Recursion Example: Towers of Hanoi • Think Recursive algorithm • Task • Move disks from left peg to right peg • When disk moved, must be placed on a peg • Only one disk (top disk on a peg) moved at a time • Larger disk may never be placed on a smaller disk

13. Recursion Example: Towers of Hanoi • Identify base case:If there is one disk move from A to C • Inductive solution for n > 1 disks • Move topmost n – 1 disks from A to B, using C for temporary storage • Move final disk remaining on A to C • Move the n – 1 disk from B to C using A for temporary storage

14. Towers of Hanoi: Solution voidhanoi(int n, conststring& initNeedle, conststring& endNeedle, conststring& tempNeedle) { if (n == 1) cout << "move " << initNeedle << " to " << endNeedle << endl; else { hanoi(n-1,initNeedle,tempNeedle,endNeedle); cout << "move " << initNeedle << " to " << endNeedle << endl; hanoi(n-1,tempNeedle,endNeedle,initNeedle); } }

15. Towers of Hanoi: Solution string beginneedle = "A", middleneedle = "B", endneedle = "C"; hanoi(3, beginneedle, endneedle, middleneedle); The solution for n = 3 move A to C move A to B move C to B move A to C move B to A move B to C move A to C output

16. Recursion Example: Towers of Hanoi • Note the graphical steps to the solution

17. Recursion Example: Parsing • Examples so far are direct recursion • Function calls itself directly • Indirect recursion occurs when • A function calls other functions • Some chain of function calls eventually results in a call to original function again • An example of this is the problem of processing arithmetic expressions

18. Recursion Example: Parsing • Parser is part of the compiler • Input to a compiler is characters • Broken up into meaningful groups • Identifiers, reserved words, constants, operators • These units are called tokens • Recognized by lexical analyzer • Syntax rules applied

19. Recursion Example: Parsing • Parser generates a parse tree using the tokens according torules below: • An expression:term + term | term – term | term • A term:factor * factor | factor / factor | factor • A factor:( expression ) | letter | digit Note the indirect recursion

20. Implementing Recursion:The Runtime Stack • Activation record created for each function call • Activation records placed on run-time stack • Recursive calls generate stack of similar activation records

21. Implementing Recursion:The Runtime Stack • When base case reached and successive calls resolved • Activation records are popped off the stack