The function description:
int ackermannFun(int l, int m)
{ if (l == 0) {
return m + 1;
} else if (m == 0)
{ return ackermannFun(l - 1, 1);
}
else
{ return ackermannFun(l - 1, ackermannFun(l, m - 1));
}
}
The base cases:
- If l is equal to zero, the method returns m + 1 .
- If m is 0, the function executes a recursive call, decrementing l by 1 and setting m to 1.
If both l and m are non-zero, the function executes a recursive function call. In this call, l is reduced by one, and m is assigned the output of the preceding recursive call where both l and m were decreased by one.
The Ackermann function is defined as:
A(a,b) = b+1; if a==0
A(a,b) =A(a-1,1); if a>0 and b==0
A(a,b) = A(a-1,A(a,b-1) ); if if a > 0 and b > 0
The Ackermann function is famous for its rapid growth rate, particularly evident even with modest inputs. As the values of a and b increase, the frequency of recursive calls also escalates, elevating the complexity of the computation. This escalation poses a challenge when estimating the function for larger a and b values.
Algorithm:
AckermannFun(a, b)
{
nextA and goalA are arrays indexed from 0 to a, with next[0] through next[m] being 0 and goal[0] through goal[a - l] being 1, and goal[a] being -1.
}
repeat
val ← nextA[0]+ 1
transfer ← true
currentValue ← O
while transfer does begin
if nextA[currentValue] = goalA[currentValue] then goalA[currentValue] ← val
else transfer ← false
nextA[currentValue] ← next[AcurrentValue]+l
currentValue ← currentValue+ 1
end while
until nextA[a] = b+ 1
return val {the value of Ack(a, b)}
end AckermannFun
This algorithm's Analysis:
- This approach has a time complexity of O(mA(a, b)) for calculating A(a, b) .
- This approach has a space complexity of O(a) to compute A(a, b) .
Example:
Let's consider an example to demonstrate the Ackermann Function in the C programming language.
// Program to implement the Ackermann function in C
#include <stdio.h>
int ackFun(int a, int b)
{
if (a== 0){
return b+1;
}
else if((a > 0) && (b== 0)){
return ackFun(a-1, 1);
}
else if((a > 0) && (b > 0)){
return ackFun(a-1, ackFun(a, b-1));
}
}
int main(){
int value;
value = ackFun(6, 4);
printf("%d", value);
return 0;
}
Explanation:
- ackFun(int a, int b). This function computes the Ackermann function iteratively.
- It complies with the definition of the Ackermann function, which includes three basic cases: If a = 0, it results in b + 1. If an is larger than 0 and b is equal to 0, the process repeats with a-1 and 1. If a and b are both larger than 0, the program recurses with a-1 and the result of a different Ackermann call with a and b-1.
- For some instances, this function does not have a return statement, which might result in undefined behavior. It must provide a value for each potential route.
- main: Calls ackFun(6, 4) to compute the Ackermann value with a = 6 and b = 4.
- The result is saved as a variable value.
- Prints the number using the printf function.
- If a = 0, it results in b + 1.
- If an is larger than 0 and b is equal to 0, the process repeats with a-1 and 1.
- If a and b are both larger than 0, the program recurses with a-1 and the result of a different Ackermann call with a and b-1.