Difference between revisions of "Recursive Functions"
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Try finding f(10). No doubt, you have the correct answer, because you intuitively stopped when you reach $f(1)$ and $f(0)$.  Try finding f(10). No doubt, you have the correct answer, because you intuitively stopped when you reach $f(1)$ and $f(0)$.  
To be formal about this, we need to define when the recursion stops, called the ''base cases''.  To be formal about this, we need to define when the recursion stops, called the ''base cases''.  
The base cases for the Fibonacci function is $f(0)=  The base cases for the Fibonacci function is $f(0)=0$, and $f(1)=1$. The typical way to write this function is as follows:  
$$f(N)=\cases{  $$f(N)=\cases{N & if $N \le 1$\cr  
f(N1)+f(N2) & if $N > 1$}$$  f(N1)+f(N2) & if $N > 1$}$$  
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::<syntaxhighlight lang="python">  ::<syntaxhighlight lang="python">  
def Fibonacci(x):  def Fibonacci(x):  
if (x =  if (x <= 1) return x  
return Fibonacci(x1) + Fibonacci(x2)  return Fibonacci(x1) + Fibonacci(x2)  
</syntaxhighlight>  </syntaxhighlight>  
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=== Factorial Function ===  === Factorial Function ===  
Consider the factorial function, $n! =  Consider the factorial function, $n! = n * (n1) * ... * 1$, with 0! defined as having a value of 1. We can define this recursively as follows:  
$$f(x)=\cases  $$f(x)=\cases  
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}$$  }$$  
With this definition, the factorial of every nonnegative integer can be found.  
Here is a Python implementation of the factorial function:  Here is a Python implementation of the factorial function:  
::<syntaxhighlight lang="python">def Factorial(x):  ::<syntaxhighlight lang="python">def Factorial(x):  
if (x  if (x == 0) return 1  
return x*Factorial(x1)  return x * Factorial(x1)  
</syntaxhighlight>  </syntaxhighlight>  
=== Some Definitions ===  
A few definitions: ''Indirection recursion'' is when a function calls another function which eventually calls the original function. For example, A calls B, and then, before function B exits, function A is called (either by B or by a function that B calls). ''Single recursion'' is recursion with a single reference to itself, such as the factorial example above. ''Multiple recursion'', illustrated by the Fibonacci number function, is when a function has multiple self references. ''Infinite recursion'' is a recursive function that never returns because it keeps calling itself. The program will eventually crash with an ''out of memory'' error message of some sort.  
This ACSL category focuses on mathematical recursive functions rather than programming algorithms. While many mathematical functions can be done iteratively more efficiently than they can be done recursively, many algorithms in computer science must be written recursively. The Tower of Hanoi which is referred to in one of the videos is a good example of this. Another example is given in the sample problems below.  
== Sample Problems==  
This ACSL category focuses on mathematical recursive functions rather than programming procedures; but you’ll see some of the latter.  This ACSL category focuses on mathematical recursive functions rather than programming procedures; but you’ll see some of the latter.  
You will typically be asked to evaluate a recursive function for some specific value.  
It's important that you are careful and neat. Work your way down the recursive calls until there are no more calls, and then work your way back up.  
=== Sample Problem 1 ===  
'''Problem:''' Find $g(11)$ given the following  
$$g(x)=\cases  
{g(x3)+1 & if $x>0$\cr  
3x & otherwise\cr  
}$$  
'''Solution:'''  
The evaluation proceeds topdown as follows:  
$$\begin{align}  
g(11)&=g(8)+1\cr  
g(8)&=g(5)+1\cr  
g(5)&=g(2)+1\cr  
g(2)&=g(1)+1\cr  
g(1)&=3\cr  
\end{align}$$  
We now use what we know about $g(1)$ to learn more values, working our way back "up" the recursion:  
:We now know the value of $g(2): g(2)=3+1=2$  
:We now know the value of $g(5): g(5)= 2+1=1$  
:We now know the value of $g(8): g(8) = 1+1=0$  
:And finally, we now have the value of $g(11): g(11)=0+1=1$  
=== Sample Problem 2 ===  
'''Problem:''' Find the value of $h(13)$ given the following definition of $h$:  
$$h(x)=\cases  
{h(x7)+1 & when $x>5$\cr  
x & when $0 \le x \le 5$\cr  
h(x+3) & when $x \lt 0$}$$  
'''Solution:'''  
$$\begin{align}  
h(13) &= h(6)+1 & \text{top rule, since $x>5$}\cr  
h(6) &= h(1)+1 & \text{top rule, since $x>5$}\cr  
h(1)&= h(1+3) =h(2)& \text{bottom rule, since $x<0$}\cr  
h(2)&=2&\text{middle rule, since $0 \le x \le 5$}\cr  
\end{align}$$  
We now work our way back up the recursion, filling in values that have been computed.  
== Sample  :$h(1) = h(2) = 2$  
:$h(6)=h(1)+1 = 2+1=3$  
:$h(13)=h(6)+1= 3+1=4$  
=== Sample Problem 3 ===  
'''Problem:''' Find the value of $f(12,6)$ given the following definition of $f$:  
$$f(x,y)=\cases  
{f(xy,y1)+2 & when $x>y$\cr  
x+y & otherwise}$$  
'''Solution:'''  
$$\begin{align}  
f(12,6) &= f(6,5)+2 & \text{top rule, since $x>y$}\cr  
f(6,5) &= f(1,4)+2 & \text{top rule, since $x>y$}\cr  
f(1,4) &= 1+4 = 5& \text{bottom rule, since $x \le y$}\cr  
\end{align}$$  
We now work our way back up the recursion, filling in values that have been computed.  
:$f(12,6) = f(6,5) + 2 = f(1,4) + 2 + 2 = 5 + 2 + 2 = 9$  
=== Sample Problem 4 ===  
'''Problem:''' Consider the following recursive algorithm for painting a square:  
1. Given a square.  
2. If the length of a side is less than 2 feet, then stop.  
3. Divide the square into 4 equal size squares (i.e., draw a “plus” sign inside the square).  
4. Paint one of these 4 small squares.  
5. Repeat this procedure (start at step 1) for each of the 3 unpainted squares.  
If this algorithm is applied to a square with a side of 16 feet (having a total area of 256 sq. feet), how many square feet will be painted?  
'''Solution:'''  
In the first pass, we get four squares of side 8.  
One is painted, three are unpainted.  
Next, we have 3*4 squares of side 4.  
Three are painted (area=3*4<sup>2</sup>), nine are not.  
Next, we have 9*4 squares of side 2.  
Nine are painted (area = 9*2<sup>2</sup>), 27 are not.  
Finally, we have 27*4 squares of side 1.  
Twentyseven are painted.  
Therefore, the total painted is 1*8<sup>2</sup> + 3*4<sup>2</sup> + 9*2<sup>2</sup> + 27*1<sup>2</sup> = 175.  
== Online Resources ==  == Online Resources ==  
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{  {  
  
 <youtube width="300" height="180">https://youtu.be/YinptBVZNFM</youtube>  
 [https://youtu.be/YinptBVZNFM "ACSL Math: Recursive Functions" (Quick Coding Bytes)]  
This video introduces the topic, then using an example problem, explains the methodology to solve problems that appear on ACSL contests.  
    
 <youtube width="300" height="180">https://youtu.be/OjZSIXStSr8</youtube>   <youtube width="300" height="180">https://youtu.be/OjZSIXStSr8</youtube>  
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The video walks through the solution to a 2variable recursive function, that is, $f(x,y)=\cases{....}$ .  The video walks through the solution to a 2variable recursive function, that is, $f(x,y)=\cases{....}$ .  
}  }  
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===Other Videos===  ===Other Videos===  
Because recursion is such a fundamental concept in computer science, there is no end to the number of YouTube videos that cover the topic. There are dozens of walkthroughs of the factorial and Fibonacci functions, and of other classic algorithms such as Towers of Hanoi and Binary Trees.  
Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.  
{  
  
 <youtube width="300" height="180">https://youtu.be/LdNdfPhwCP4</youtube>  
 [https://youtu.be/LdNdfPhwCP4 ''Recursion lecture 1'' ('''anim aland''')]  
Prof. Alan Dorin from Monash University presents an wonderful introduction to recursion. The last part of the video uses factorial as an example. Taken note how his base case guards against infinite recursion.  
  
 <youtube width="300" height="180">https://youtu.be/dxyYP3BSdcQ</youtube>  
 [https://youtu.be/dxyYP3BSdcQ ''Fibonacci Sequence  Anatomy of recursion and space complexity analysis'' ('''mycodeschool''')]  
The video is handsimulation on the whiteboard of the recursive code for computing a Fibonnaci number. The code uses a base case of <syntaxhighlight lang="python" inline>if n<=1 return n</syntaxhighlight> to prevent infinite recursion where the function called with a negative parameter.  
} 
Latest revision as of 21:18, 7 December 2021
A definition that defines an object in terms of itself is said to be recursive. In computer science, recursion refers to a function or subroutine that calls itself, and it is a fundamental paradigm in programming. A recursive program is used for solving problems that can be broken down into subproblems of the same type, doing so until the problem is easy enough to solve directly.
Examples
Fibonacci Numbers
A common recursive function that you’ve probably encountered is the Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, and so on. That is, you get the next Fibonacci number by adding together the previous two. Mathematically, this is written as
$$f(N)=f(N1)+f(N2)$$
Try finding f(10). No doubt, you have the correct answer, because you intuitively stopped when you reach $f(1)$ and $f(0)$. To be formal about this, we need to define when the recursion stops, called the base cases. The base cases for the Fibonacci function is $f(0)=0$, and $f(1)=1$. The typical way to write this function is as follows: $$f(N)=\cases{N & if $N \le 1$\cr f(N1)+f(N2) & if $N > 1$}$$
Here is a Python implementation of the Fibonacci function:
def Fibonacci(x): if (x <= 1) return x return Fibonacci(x1) + Fibonacci(x2)
(As a challenge to the reader: How could you implement the Fibonacci function without using recursion?)
Factorial Function
Consider the factorial function, $n! = n * (n1) * ... * 1$, with 0! defined as having a value of 1. We can define this recursively as follows:
$$f(x)=\cases {1 & if $x=0$\cr x*f(x1) & if $x\gt 0$\cr }$$
With this definition, the factorial of every nonnegative integer can be found.
Here is a Python implementation of the factorial function:
def Factorial(x): if (x == 0) return 1 return x * Factorial(x1)
Some Definitions
A few definitions: Indirection recursion is when a function calls another function which eventually calls the original function. For example, A calls B, and then, before function B exits, function A is called (either by B or by a function that B calls). Single recursion is recursion with a single reference to itself, such as the factorial example above. Multiple recursion, illustrated by the Fibonacci number function, is when a function has multiple self references. Infinite recursion is a recursive function that never returns because it keeps calling itself. The program will eventually crash with an out of memory error message of some sort.
This ACSL category focuses on mathematical recursive functions rather than programming algorithms. While many mathematical functions can be done iteratively more efficiently than they can be done recursively, many algorithms in computer science must be written recursively. The Tower of Hanoi which is referred to in one of the videos is a good example of this. Another example is given in the sample problems below.
Sample Problems
This ACSL category focuses on mathematical recursive functions rather than programming procedures; but you’ll see some of the latter. You will typically be asked to evaluate a recursive function for some specific value. It's important that you are careful and neat. Work your way down the recursive calls until there are no more calls, and then work your way back up.
Sample Problem 1
Problem: Find $g(11)$ given the following $$g(x)=\cases {g(x3)+1 & if $x>0$\cr 3x & otherwise\cr }$$
Solution:
The evaluation proceeds topdown as follows:
$$\begin{align} g(11)&=g(8)+1\cr g(8)&=g(5)+1\cr g(5)&=g(2)+1\cr g(2)&=g(1)+1\cr g(1)&=3\cr \end{align}$$
We now use what we know about $g(1)$ to learn more values, working our way back "up" the recursion:
 We now know the value of $g(2): g(2)=3+1=2$
 We now know the value of $g(5): g(5)= 2+1=1$
 We now know the value of $g(8): g(8) = 1+1=0$
 And finally, we now have the value of $g(11): g(11)=0+1=1$
Sample Problem 2
Problem: Find the value of $h(13)$ given the following definition of $h$:
$$h(x)=\cases {h(x7)+1 & when $x>5$\cr x & when $0 \le x \le 5$\cr h(x+3) & when $x \lt 0$}$$
Solution:
$$\begin{align} h(13) &= h(6)+1 & \text{top rule, since $x>5$}\cr h(6) &= h(1)+1 & \text{top rule, since $x>5$}\cr h(1)&= h(1+3) =h(2)& \text{bottom rule, since $x<0$}\cr h(2)&=2&\text{middle rule, since $0 \le x \le 5$}\cr \end{align}$$
We now work our way back up the recursion, filling in values that have been computed.
 $h(1) = h(2) = 2$
 $h(6)=h(1)+1 = 2+1=3$
 $h(13)=h(6)+1= 3+1=4$
Sample Problem 3
Problem: Find the value of $f(12,6)$ given the following definition of $f$:
$$f(x,y)=\cases {f(xy,y1)+2 & when $x>y$\cr x+y & otherwise}$$
Solution:
$$\begin{align} f(12,6) &= f(6,5)+2 & \text{top rule, since $x>y$}\cr f(6,5) &= f(1,4)+2 & \text{top rule, since $x>y$}\cr f(1,4) &= 1+4 = 5& \text{bottom rule, since $x \le y$}\cr \end{align}$$
We now work our way back up the recursion, filling in values that have been computed.
 $f(12,6) = f(6,5) + 2 = f(1,4) + 2 + 2 = 5 + 2 + 2 = 9$
Sample Problem 4
Problem: Consider the following recursive algorithm for painting a square:
1. Given a square. 2. If the length of a side is less than 2 feet, then stop. 3. Divide the square into 4 equal size squares (i.e., draw a “plus” sign inside the square). 4. Paint one of these 4 small squares. 5. Repeat this procedure (start at step 1) for each of the 3 unpainted squares.
If this algorithm is applied to a square with a side of 16 feet (having a total area of 256 sq. feet), how many square feet will be painted?
Solution:
In the first pass, we get four squares of side 8. One is painted, three are unpainted. Next, we have 3*4 squares of side 4. Three are painted (area=3*4^{2}), nine are not. Next, we have 9*4 squares of side 2. Nine are painted (area = 9*2^{2}), 27 are not. Finally, we have 27*4 squares of side 1. Twentyseven are painted. Therefore, the total painted is 1*8^{2} + 3*4^{2} + 9*2^{2} + 27*1^{2} = 175.
Online Resources
ACSL
The following videos show the solution to problems that have appeared in previous ACSL contests.
"ACSL Math: Recursive Functions" (Quick Coding Bytes)
This video introduces the topic, then using an example problem, explains the methodology to solve problems that appear on ACSL contests.  
Recursion Example 1 (CalculusNguyenify)
The video walks through the solution to a straightforward singlevariable recursive function, that is, $f(x)=\cases{....}$ The problem appeared in ACSL Senior Division Contest #1, 20142015.  
Recursion Example 2 (CalculusNguyenify)
The video walks through the solution to a 2variable recursive function, that is, $f(x,y)=\cases{....}$ . The problem appeared in ACSL Senior Division Contest #1, 20142015.  
Recursive Functions ACSL Example Problem (Tangerine Code)
The video walks through the solution to a 2variable recursive function, that is, $f(x,y)=\cases{....}$ .

Other Videos
Because recursion is such a fundamental concept in computer science, there is no end to the number of YouTube videos that cover the topic. There are dozens of walkthroughs of the factorial and Fibonacci functions, and of other classic algorithms such as Towers of Hanoi and Binary Trees. Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.
Recursion lecture 1 (anim aland)
Prof. Alan Dorin from Monash University presents an wonderful introduction to recursion. The last part of the video uses factorial as an example. Taken note how his base case guards against infinite recursion.  
Fibonacci Sequence  Anatomy of recursion and space complexity analysis (mycodeschool)
The video is handsimulation on the whiteboard of the recursive code for computing a Fibonnaci number. The code uses a base case of
