### Tuesday, June 27, 2006

## Recursion vs Iteration

When writing code to do repetitive tasks, the two primary approaches are iteration and recursion. Generally you can use either one interchangeably, but potentially with different performance and complexity.

A

CREATE OR REPLACE FUNCTION FIBONACCI_REC (in_number IN NUMBER)

RETURN NUMBER DETERMINISTIC

AS

BEGIN

CASE

WHEN in_number = 1 THEN RETURN 1;

WHEN in_number = 2 THEN RETURN 1;

ELSE RETURN (FIBONACCI_REC (in_number - 1) + FIBONACCI_REC (in_number - 2));

END CASE;

END;

An

CREATE OR REPLACE FUNCTION FIBONACCI_IT (in_number IN NUMBER)

RETURN NUMBER DETERMINISTIC

AS

fib1 NUMBER := 1;

fib2 NUMBER := 1;

fib3 NUMBER := 1;

fib4 NUMBER;

BEGIN

FOR fib4 IN 3 .. in_number

LOOP

fib3 := fib1 + fib2;

fib1 := fib2;

fib2 := fib3;

END LOOP;

RETURN fib3;

END;

The advantages and disadvantages of the two are not always obvious, and you should really take it on a case-by-case basis. When in doubt: test it. But generally:

1. Recursion may be slower, and use greater resources, because of the extra function calls.

2. Recursion may lead to simpler, shorter, easier-to-understand functions, especially for mathematicians who are comfortable with induction formulae.

Either way, one of the most critical aspects of writing either a recursive or an iterative function is to have an exit clause (or "base case" for recursions) that is checked at every recursion/iteration and is guaranteed to be reached at some finite point, given any input. Otherwise you will get either:

I can illustrate this point with my recursive example above. Do NOT try this, but if you inserted a negative number, what do you think would happen? Infinite recursion.

Now you have not only the basics on iteration and recursion, but, based on the topic I chose for the sample code, knowledge that I saw "Da Vinci Code" recently.

A

**recursive**function calls itself (possibly more than once), with different parameters, and defines an exit clause that is guaranteed to be reached.CREATE OR REPLACE FUNCTION FIBONACCI_REC (in_number IN NUMBER)

RETURN NUMBER DETERMINISTIC

AS

BEGIN

CASE

WHEN in_number = 1 THEN RETURN 1;

WHEN in_number = 2 THEN RETURN 1;

ELSE RETURN (FIBONACCI_REC (in_number - 1) + FIBONACCI_REC (in_number - 2));

END CASE;

END;

An

**iterative**function includes a loop, which iterates a pre-determined number of times, or checks for an exit clause every time through.CREATE OR REPLACE FUNCTION FIBONACCI_IT (in_number IN NUMBER)

RETURN NUMBER DETERMINISTIC

AS

fib1 NUMBER := 1;

fib2 NUMBER := 1;

fib3 NUMBER := 1;

fib4 NUMBER;

BEGIN

FOR fib4 IN 3 .. in_number

LOOP

fib3 := fib1 + fib2;

fib1 := fib2;

fib2 := fib3;

END LOOP;

RETURN fib3;

END;

The advantages and disadvantages of the two are not always obvious, and you should really take it on a case-by-case basis. When in doubt: test it. But generally:

1. Recursion may be slower, and use greater resources, because of the extra function calls.

2. Recursion may lead to simpler, shorter, easier-to-understand functions, especially for mathematicians who are comfortable with induction formulae.

Either way, one of the most critical aspects of writing either a recursive or an iterative function is to have an exit clause (or "base case" for recursions) that is checked at every recursion/iteration and is guaranteed to be reached at some finite point, given any input. Otherwise you will get either:

**Infinite Recursion**: More and more functions being spawned, never closing, using up resources until there are none left, possibly crashing the system.**Infinite loop**: A loop that cycles forever, burning CPU and never completing.I can illustrate this point with my recursive example above. Do NOT try this, but if you inserted a negative number, what do you think would happen? Infinite recursion.

Now you have not only the basics on iteration and recursion, but, based on the topic I chose for the sample code, knowledge that I saw "Da Vinci Code" recently.

Comments:

Did you learn that from experience? ;-)

LewisC

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*Do NOT try this, but if you inserted a negative number, what do you think would happen?*

Did you learn that from experience? ;-)

LewisC

You are right, recursive is much more slower...

SQL> SELECT fibonacci_rec(35) FROM dual;

FIBONACCI_REC(35)

-----------------

9227465

Executed in 50,963 seconds

SQL> SELECT fibonacci_it(35) FROM dual;

FIBONACCI_IT(35)

----------------

9227465

Executed in 0,07 seconds

SQL>

SQL> SELECT fibonacci_rec(35) FROM dual;

FIBONACCI_REC(35)

-----------------

9227465

Executed in 50,963 seconds

SQL> SELECT fibonacci_it(35) FROM dual;

FIBONACCI_IT(35)

----------------

9227465

Executed in 0,07 seconds

SQL>

Your benchmark is wrong or at least very confusing.

You are presenting two versions of Fibonacci and you are claming that the performance difference is due to "the extra function calls". That's very far from being a complete explanation and it can be very confusing for people reading this article.

The main difference is algorithmical. The recursion uses a trivial exponential algorithm (because the recursive calls are overlaping), whereas the iterative one uses a dynamic programing altorithm with linear execution time.

Btw. : Fibonacci is the first example of pure algorithmic gain in a dynamic programing course.

In order to build a correct benchmark you must

- either chose a case where recursive and iterative versions have the same time complexity (say linear). For example, use the sum of the first n integers.

- or explain that the poor performance of the recursive function from your example come from the huge algorithmic difference and not from the lack of performance of the recursive calls.

You should really update this post, because as it is it represents a pure disinformation.

You are presenting two versions of Fibonacci and you are claming that the performance difference is due to "the extra function calls". That's very far from being a complete explanation and it can be very confusing for people reading this article.

The main difference is algorithmical. The recursion uses a trivial exponential algorithm (because the recursive calls are overlaping), whereas the iterative one uses a dynamic programing altorithm with linear execution time.

Btw. : Fibonacci is the first example of pure algorithmic gain in a dynamic programing course.

In order to build a correct benchmark you must

- either chose a case where recursive and iterative versions have the same time complexity (say linear). For example, use the sum of the first n integers.

- or explain that the poor performance of the recursive function from your example come from the huge algorithmic difference and not from the lack of performance of the recursive calls.

You should really update this post, because as it is it represents a pure disinformation.

"Pure disinformation?"

I make many points, you feel one of my suggestions is correct but unclear, and you jump to "pure disinformation?" :)

Anyway, thanks for that clarification. The examples were provided to show how the same problem is solved using the different methods, not as benchmarks. But the better performance of the iterative function is a direct result of using iteration. You could never find as good an "algorithmic" solution using recursion: every time you'd get more recursive function calls than iterative loops.

I think this just goes to show that:

"you should really take it on a case-by-case basis. When in doubt: test it."

I make many points, you feel one of my suggestions is correct but unclear, and you jump to "pure disinformation?" :)

Anyway, thanks for that clarification. The examples were provided to show how the same problem is solved using the different methods, not as benchmarks. But the better performance of the iterative function is a direct result of using iteration. You could never find as good an "algorithmic" solution using recursion: every time you'd get more recursive function calls than iterative loops.

I think this just goes to show that:

"you should really take it on a case-by-case basis. When in doubt: test it."

> I make many points, you feel one of my suggestions is

> correct but unclear, and you jump to "pure disinformation?" :)

you are right : I should have used the word misinformation instead. Sorry about that

> I think this just goes to show that:

> "you should really take it on a case-by-case basis. When in doubt: test it."

I would say better : "When in doubt: think, and if you are still in doubt, than test it". The simple reason being that sometimes thinking can go faster than testing.

:-)

> correct but unclear, and you jump to "pure disinformation?" :)

you are right : I should have used the word misinformation instead. Sorry about that

> I think this just goes to show that:

> "you should really take it on a case-by-case basis. When in doubt: test it."

I would say better : "When in doubt: think, and if you are still in doubt, than test it". The simple reason being that sometimes thinking can go faster than testing.

:-)

> You could never find as good an "algorithmic" solution using recursion:

> every time you'd get more recursive function calls than iterative loops

yes you can do it : use a lookup in an array (or hash table) before doing the recursive call. the size of the array is the maximum number of Fibonacci we can compute. In this case, you have n-1 calls, instead of 2^n calls (quite a difference, right ? ;-)).

something like that (Java - C# like syntax, hope it's OK for you) :

private int[] fibonacci;

public Fibonacci() {

// this could be replaced with lazy initialization too, but there would be an additional overhead

fibonacci = new int[MAX_FIBONACCI_NUMBER];

fibonacci[0] = 1;

fibonacci[1] = 1;

}

public int FibonacciRecLookup(int n) {

int fib1 = fibonacci[n-1] != 0 ? fibonacci[n-1] : FibonacciRecLookup(n-1);

int fib2 = fibonacci[n-2] != 0 ? fibonacci[n-2] : FibonacciRecLookup(n-2);

fibonacci[n] = fib1 + fib2;

return fibonacci[n];

}

Of course, in this case we have the same number of function calls as iterative loops, but there's another overhead : the extra lookups. But we still have a quite simple code, similar to the recursive one.

This can be used as a general technique to obtain the optimal tradeof between performance and code simplicity.

> every time you'd get more recursive function calls than iterative loops

yes you can do it : use a lookup in an array (or hash table) before doing the recursive call. the size of the array is the maximum number of Fibonacci we can compute. In this case, you have n-1 calls, instead of 2^n calls (quite a difference, right ? ;-)).

something like that (Java - C# like syntax, hope it's OK for you) :

private int[] fibonacci;

public Fibonacci() {

// this could be replaced with lazy initialization too, but there would be an additional overhead

fibonacci = new int[MAX_FIBONACCI_NUMBER];

fibonacci[0] = 1;

fibonacci[1] = 1;

}

public int FibonacciRecLookup(int n) {

int fib1 = fibonacci[n-1] != 0 ? fibonacci[n-1] : FibonacciRecLookup(n-1);

int fib2 = fibonacci[n-2] != 0 ? fibonacci[n-2] : FibonacciRecLookup(n-2);

fibonacci[n] = fib1 + fib2;

return fibonacci[n];

}

Of course, in this case we have the same number of function calls as iterative loops, but there's another overhead : the extra lookups. But we still have a quite simple code, similar to the recursive one.

This can be used as a general technique to obtain the optimal tradeof between performance and code simplicity.

"Misinformation" - Yes, who said Rumanians had no sense of humour?

Do you believe that your provided example is true recursion? Even if so, you seem to be suggesting that you must use programming techniques to make recursion almost as fast as iteration. Which, in the end, is still an agreement with my initial assertions.

P.S. Testing is more reliable than thinking.

Do you believe that your provided example is true recursion? Even if so, you seem to be suggesting that you must use programming techniques to make recursion almost as fast as iteration. Which, in the end, is still an agreement with my initial assertions.

P.S. Testing is more reliable than thinking.

> Do you believe that your provided example is true recursion?

absolutely, an intelligent recursion.

> you seem to be suggesting that you must use programming techniques to make recursion almost as fast as iteration

I'm only suggesting that

1. when you want to compare recursive and iterative algorithms, the only reliable way to get a fair comparison is to look at the *best* recursive and at the *best* iterative algorithms. Unfortunately you made a comparison between the worst possible recursive algorithm and a good iterative one. So your comparison can not lead to good conclusions regarding the "function call overhead".

2. When implementing recursion, one should take into account that there are many ways to do it

> Which, in the end, is still an agreement with my initial assertions.

in your dreams :-)

> P.S. Testing is more reliable than thinking.

that's not always true. For instance, in your example, one can see immediately that the recursion performs in time 2^n (exponential) whereas the iteration performs in time n (linear). Once you get that, testing is useless (it takes longer and have no added value, it only confirms an obvious difference). If there is no strong evidence of the algorithmic difference, testing become more reliable.

and please note that I am not speaking of unit or functional testing, but of performance testing. Unit ant functional are obviously more reliable than thinking. But when doing performance testing, algorithmic thinking can be a valuable tool.

absolutely, an intelligent recursion.

> you seem to be suggesting that you must use programming techniques to make recursion almost as fast as iteration

I'm only suggesting that

1. when you want to compare recursive and iterative algorithms, the only reliable way to get a fair comparison is to look at the *best* recursive and at the *best* iterative algorithms. Unfortunately you made a comparison between the worst possible recursive algorithm and a good iterative one. So your comparison can not lead to good conclusions regarding the "function call overhead".

2. When implementing recursion, one should take into account that there are many ways to do it

> Which, in the end, is still an agreement with my initial assertions.

in your dreams :-)

> P.S. Testing is more reliable than thinking.

that's not always true. For instance, in your example, one can see immediately that the recursion performs in time 2^n (exponential) whereas the iteration performs in time n (linear). Once you get that, testing is useless (it takes longer and have no added value, it only confirms an obvious difference). If there is no strong evidence of the algorithmic difference, testing become more reliable.

and please note that I am not speaking of unit or functional testing, but of performance testing. Unit ant functional are obviously more reliable than thinking. But when doing performance testing, algorithmic thinking can be a valuable tool.

Now that we're talking about procedure call overhead ...

http://thinkoracle.blogspot.com/2006/08/plsql-procedure-call-overhead-re.html

(Hint: It's negligible)

http://thinkoracle.blogspot.com/2006/08/plsql-procedure-call-overhead-re.html

(Hint: It's negligible)

A bit late, but usually it's even faster if you rewrite the recursive function... For fibonacci you can find such a function at (eg:) http://mathforum.org/library/drmath/view/52686.html

Notice that this function does not require you to calculate the previous n-1 results...

Notice that this function does not require you to calculate the previous n-1 results...

I am smart, and I have data structure today. and I LOVE IT!

data storage is sexy once u get to feel it..

data storage is sexy once u get to feel it..

I try to compute or to google closed-form first, when I'm computing anything like fibonacci..

complexity is then O(1)

http://en.wikipedia.org/wiki/Fibonacci_number#Closed_form_expression

complexity is then O(1)

http://en.wikipedia.org/wiki/Fibonacci_number#Closed_form_expression

This is such a nice post, actually I'am new in Oracle industry and I don't know much about so that's why I'am looking on the web for tutorials and Oracle Consulting and Oracle Services to gain my knowledge about Oracle. Thank you so much for sharing such a nice post with us.

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