When to use recursion

Computer Science 4

Mr. Gerb

Reference:

Objective: Understand when recursion is useful and how to remove it.

Three rules of thumb: Use recursion only when:

• The depth of recursive calls is relatively “shallow” compared to the size of the problem.

• The recursive version does about the same amount of work as the nonrecursive version.

• The recursive version is shorter and simpler than the nonrecursive solution.

Shallow recursive calls: First Example

• Supposed you were to use recursion to search a very large linked list with N elements.

• O(N) stack frames might exist at one time– Bad if N is large– Most compilers have a limited amount of space

set aside for the stack– Likely to cause a stack overflow

Shallow Recursive Calls: Second Example

• Suppose you were to use recursion to binary search an array of N elements

• O(LogN) stack frames will exist at one time– OK even if N is large– E.g. for 16 billion elements, only need 34 stack

frames!– I.e. depth of recursion not a problem for binary

search

Recursive Version Works Same as Non-Recursive Version

• Example: Nth fibonacci number:

public static int fib(int n){

if (n<=2)

return 1;

else

return fib(n-1)+fib(n-2);

}• How many recursive calls does fib make?

Fibonacci Example Cont’d

• fib(1) and fib(2) both take none• fib(3) takes two:

fib(1)fib(2)

• fib(4) takes 4fib(3) fib(1) fib(2) fib(2)

• fib(5) takes 8fib(4) fib(3) fib(1) fib(2)fib(2) fib(3) fib(1) fib(2)

Fibonacci Example Cont’d• fib(6) takes 14

fib(5) fib(4) fib(3) fib(1) fib(2) fib(2) fib(3) fib(1) fib(2) fib(4) fib(3) fib(1) fib(2) fib(2)

• fib(7) takes 24fib(6) fib(5) fib(4) fib(3) fib(1) fib(2) fib(2) fib(3) fib(1) fib(2) fib(4) fib(3) fib(1) fib(2) fib(2) fib(5)fib(4) fib(3) fib(1) fib(2) fib(2) fib(3) fib(1) fib(2)

Fibonacci Example Cont’d• fib(8) takes 40

fib(6) fib(5) fib(4) fib(3) fib(1) fib(2) fib(2) fib(3) fib(1) fib(2) fib(4) fib(3) fib(1) fib(2) fib(2) fib(7)fib(6) fib(5) fib(4) fib(3) fib(1) fib(2) fib(2) fib(3) fib(1) fib(2) fib(4) fib(3) fib(1) fib(2) fib(2) fib(5)fib(4) fib(3) fib(1) fib(2) fib(2) fib(3) fib(1) fib(2)

Non-Recursive Fibonacci

public static void fib(int n){ int back1=1, back2=1,fib=1; for (int ctr=3;ctr<=n;ctr++){ fib=back1+back2; back2=back1; back1=fib; } return fib;}

Recursive vs. Non-Recursive Fibonacci

• Non-recursive Fibonacci equires n-2 runs through the loop– fib(3) once through the loop– fib(4) twice through the loop– fib(8) six times through the loop

• Recursive Fibonacci requires an exponential number of recursive calls

• Therefore computing Fibonacci numbers is not a good candidate for recursion.

Recursive Function Shorter and Simpler

• Many recursive algorithms would require creating your own stack to implement non-recursively– Merge Sort– Quick Sort– Search

• Much simpler to express, understand, implement and maintain using recursion

Two ways to remove recursion

• Use a stack– Keep your own stack frames

– Rarely an improvement on the recursive solution

• Tail recursion– When a function contains only one recursive call and it

is the last statement to be executed, that is called tail recursion

– Tail recursion can be replaced by iteration to remove recursion.

Example: Search an array for an item

if (st>a.length)

return false;

else if (a[st]==n)

return true;

else

return fnd(a,n,st+1);

}

int found = false; while (!found && st<=a.length){ if (a[st]==n) found = true; st++ }}

public boolean fnd(int[] a, int n, int st){

Summary

• Use recursion only when– Recursion is not deep– Doesn’t slow the algorithm down– Easier to understand or implement

• Can remove recursion with a stack or tail recursion

• Tail recursion– One recursive call, last statement executed– Recursion can be replaced with iteration