Ackermann’s function using Dynamic programming

Given two non-zero integers M and N, the problem is to compute the result of the Ackermann function based on some particular equations.

Ackermann function is defined as:

Examples:

Input: M = 2, N = 2
Output: 7

Input: M = 2, N = 7
Output: 6141004759

The approach for Ackermann function described in this article, takes a very huge amount of time to compute the value for even small values of (M, N) or in most cases doesn’t result in anything.

Dynamic Programming approach:

Here are the following Ackermann equations that would be used to come up with efficient solution.

A(m, n) = A(m-1, A(m, n-1)) —– (Eq 1)

A(0, n) = n+1 —– (Eq 2)

A(m, 0) = A(m-1, 1) —– (Eq 3)

Let’s assume the value of m = 2 and n = 2

A 2d DP table of size ( (m+1) x (n+1) ) is created for storing the result of each sub-problem.

Following are the steps demonstrated to fill up the table.

1. Empty Table – Initial Step

Filled using A ( 0, n ) = n + 1 The very next method is to fill all the base values, by taking the help of equation-2.

In the next step the whole 1st row would be filled,

A ( 1, 0 ) = A ( 0, 1 ) —– (refer Eq (3)) Since A ( 0, 1 ) = 2 Therefore, A ( 1, 0 ) = 2 —–(Eq 4)

A ( 1, 1 ) = A ( 0, A ( 1, 0 ) ) —– refer Eq (1) = A ( 0, 2 ) —– refer Eq (4) = 3 —– refer Eq (2) So, A ( 1, 1 ) = 3 —–(Eq 5)

A ( 1, 2 ) = A ( 0, A ( 1, 1 ) ) —– refer equation (1) = A ( 0, 3 ) —– refer equation (5) = 4 —– refer equation (2) So, A ( 1, 2 ) = 4 —–(Eq 6)

1. Fill the table using equations and stored values

Let’s just fill the first column of the last row i.e (2, 0) in the same manner as above, because for the other two columns there are some unsolved values.

A ( 2, 0 ) = A ( 1, 1 ) —– refer equation (3) A ( 1, 1 ) = 3 So, A ( 2, 0 ) = 3 —– (Eq. 7)

Solving for A ( 2, 1 ) and A ( 2, 2 ).

For simplicity the process for solving above functions is divided into two steps,

• In the first one, the problem is identified.
  

A ( 2, 1 ) = A ( 1, A ( 2, 0 ) ) —– refer equation (1) A ( 2, 0 ) = 3 A ( 2, 1 ) = A ( 1, 3 ) So to compute this value, again use equation (1) A ( 1, 3 ) = A ( 0, A ( 1, 2 ) ) A ( 1, 2 ) = 4 A ( 1, 3 ) = A ( 0, 4 ) —– refer equation(2) = 5 Therefore A ( 2, 1 ) = A ( 1, 3 ) = 5 — (Eq 7)

• In the next one, methodology is described in detail and a generic formula is obtained to be logical while being used in program

Let’s solve A ( 2, 2 ) , with a theory upfront

A ( 2, 2 ) = A ( 1, A ( 2, 1 ) ) —– refer equation (1) A ( 2, 1) = 5 A ( 2, 2 ) = A ( 1, 5 )

To compute A(1, 5) in a generic manner, observe how it reduces itself!

A ( 1, 5 ) = A ( 0, A ( 1, 4 ) ) A ( 1, 4 ) = A( 0, A ( 1, 3 ) ) A ( 1, 3 ) = A ( 0, A ( 1, 2 ) ) A ( 1, 2 ) = 4 Returning back from the function we get, A ( 1, 3 ) = A ( 0, 4 ) = 5 —– refer equation (2) A ( 1, 4 ) = A ( 0, A (1, 3 ) ) = A ( 0, 5 ) = 6 —–Since A ( 1, 3 ) = 5 A ( 1, 5 ) = A ( 0, A ( 1, 4 ) ) = A ( 0, 6 ) = 7 So, A ( 2, 2 ) = 7 ——- (Eq 9)

1. Final Table with result of each sub-problem

Below is the implementation of the above approach: 

C++

#include <iostream>
#include <string>
#include <vector>
using namespace std;

// Bottom Up Approach
long long int Ackermann(int m, int n){

    // creating 2D LIST
    vector<vector<long long int>> cache(m + 1, vector<long long int>(n + 1, 0));

    for (int rows = 0 ; rows < m + 1 ; rows++){
        for (int cols = 0 ; cols < n + 1 ; cols++){

            // base case A ( 0, n ) = n + 1
            if (rows == 0){
                cache[rows][cols] = cols + 1;
            }

            // base case A ( m, 0 ) =
            // A ( m-1, 1) [Computed already]
            else if (cols == 0)
            {
                cache[rows][cols] = cache[rows-1][1];
            }
            else
            {

                // if rows and cols > 0
                // then applying A ( m, n ) =
                // A ( m-1, A ( m, n-1 ) )
                long long int r = rows - 1;
                long long int c = cache[rows][cols-1];
                long long int ans;

                // applying equation (2)
                // here A ( 0, n ) = n + 1
                if (r == 0){
                    ans = c + 1;
                }else if (c <= n){
                    // using stored value in cache
                    ans = cache[rows-1][cache[rows][cols-1]];
                }else{
                    // Using the Derived Formula
                    // to compute mystery values in O(1) time
                    ans = (c-n)*(r) + cache[r][n];
                }

                cache[rows][cols] = ans;
            }
        }
    }
    return cache[m][n];
}


// Driver code
int main()
{

    // very small values
    int m = 2;
    int n = 2;

    // a bit higher value
    int m1 = 5;
    int n1 = 7;


    cout << "Ackermann value for m = " << m <<
      " and n = " << n << " is -> " <<
      Ackermann(m, n) << endl;

    cout << "Ackermann value for m = " <<
      m1 << " and n = " << n1 << " is -> " <<
      Ackermann(m1, n1) << endl;

    return 0;
}

// This code is contributed by subhamgoyal2014.

Java

// Java program to implement the approach
import java.util.*;
class GFG {

    // Bottom Up Approach
    static long Ackermann(int m, int n)
    {

        // creating 2D LIST
        ArrayList<ArrayList<Long> > cache
            = new ArrayList<ArrayList<Long> >();
        for (int i = 0; i <= m; i++) {
            cache.add(new ArrayList<Long>());
            for (int j = 0; j <= n; j++) {
                ArrayList<Long> l1 = cache.get(i);
                l1.add(0L);
                cache.set(i, l1);
            }
        }

        for (int rows = 0; rows < m + 1; rows++) {
            for (int cols = 0; cols < n + 1; cols++) {

                // base case A ( 0, n ) = n + 1
                if (rows == 0) {
                    ArrayList<Long> l1 = cache.get(rows);
                    l1.set(cols, cols + 1L);
                    cache.set(rows, l1);
                }

                // base case A ( m, 0 ) =
                // A ( m-1, 1) [Computed already]
                else if (cols == 0) {
                    ArrayList<Long> l1 = cache.get(rows);
                    l1.set(cols,
                           cache.get(rows - 1).get(1));
                    cache.set(rows, l1);
                }
                else {

                    // if rows and cols > 0
                    // then applying A ( m, n ) =
                    // A ( m-1, A ( m, n-1 ) )
                    long r = rows - 1;
                    long c = cache.get(rows).get(cols - 1);
                    long ans;

                    // applying equation (2)
                    // here A ( 0, n ) = n + 1
                    if (r == 0) {
                        ans = c + 1;
                    }
                    else if (c <= n) {
                        // using stored value in cache
                        ans = cache.get(rows - 1).get(
                            Math.toIntExact(
                                cache.get(rows).get(
                                    (int)cols - 1)));
                    }
                    else {
                        // Using the Derived Formula
                        // to compute mystery values in O(1)
                        // time
                        ans = (c - n) * (r)
                              + cache.get((int)r).get(n);
                    }
                    ArrayList<Long> l1 = cache.get(rows);
                    l1.set(cols, ans);
                    cache.set(rows, l1);
                }
            }
        }
        return cache.get(m).get(n);
    }

    // Driver code
    public static void main(String[] args)
    {

        // very small values
        int m = 2;
        int n = 2;

        // a bit higher value
        int m1 = 5;
        int n1 = 7;

        System.out.println("Ackermann value for m = " + m
                           + " and n = " + n + " is -> "
                           + Ackermann(m, n));

        System.out.println("Ackermann value for m = " + m1
                           + " and n = " + n1 + " is -> "
                           + Ackermann(m1, n1));
    }
}

// This code is contributed by phasing17.

Python3

# Python code for the above approach

# Bottom Up Approach


def Ackermann(m, n):

    # creating 2D LIST
    cache = [[0 for i in range(n + 1)] for j in range(m + 1)]
    for rows in range(m + 1):
        for cols in range(n + 1):
            # base case A ( 0, n ) = n + 1
            if rows == 0:
                cache[rows][cols] = cols + 1
            # base case  A ( m, 0 ) =
            # A ( m-1, 1) [Computed already]
            elif cols == 0:
                cache[rows][cols] = cache[rows-1][1]
            else:
                # if rows and cols > 0
                # then applying A ( m, n ) =
                # A ( m-1, A ( m, n-1 ) )
                r = rows - 1
                c = cache[rows][cols-1]
                # applying equation (2)
                # here A ( 0, n ) = n + 1
                if r == 0:
                    ans = c + 1
                elif c <= n:
                    # using stored value in cache
                    ans = cache[rows-1][cache[rows][cols-1]]
                else:
                    # Using the Derived Formula
                    # to compute mystery values in O(1) time
                    ans = (c-n)*(r) + cache[r][n]

                cache[rows][cols] = ans

    return cache[m][n]


# very small values
m = 2
n = 2

# a bit higher value
m1 = 5
n1 = 7


print("Ackermann value for m = ", m,
      " and n = ", n, "is -> ",
      Ackermann(m, n))

print("Ackermann value for m = ", m1,
      " and n = ", n1, "is -> ",
      Ackermann(m1, n1))

C#

// C# program to implement the approach

using System;
using System.Collections.Generic;

class GFG {
    // Bottom Up Approach
    static long Ackermann(int m, int n)
    {

        // creating 2D LIST
        List<List<long> > cache = new List<List<long> >();
        for (int i = 0; i <= m; i++) {
            cache.Add(new List<long>());
            for (int j = 0; j <= n; j++)
                cache[i].Add(0);
        }

        for (int rows = 0; rows < m + 1; rows++) {
            for (int cols = 0; cols < n + 1; cols++) {

                // base case A ( 0, n ) = n + 1
                if (rows == 0) {
                    cache[rows][cols] = cols + 1;
                }

                // base case A ( m, 0 ) =
                // A ( m-1, 1) [Computed already]
                else if (cols == 0) {
                    cache[rows][cols] = cache[rows - 1][1];
                }
                else {

                    // if rows and cols > 0
                    // then applying A ( m, n ) =
                    // A ( m-1, A ( m, n-1 ) )
                    long r = rows - 1;
                    long c = cache[rows][cols - 1];
                    long ans;

                    // applying equation (2)
                    // here A ( 0, n ) = n + 1
                    if (r == 0) {
                        ans = c + 1;
                    }
                    else if (c <= n) {
                        // using stored value in cache
                        ans = cache[rows - 1][(
                            int)cache[rows][cols - 1]];
                    }
                    else {
                        // Using the Derived Formula
                        // to compute mystery values in O(1)
                        // time
                        ans = (c - n) * (r)
                              + cache[(int)r][n];
                    }

                    cache[rows][cols] = ans;
                }
            }
        }
        return cache[m][n];
    }

    // Driver code
    public static void Main(string[] args)
    {

        // very small values
        int m = 2;
        int n = 2;

        // a bit higher value
        int m1 = 5;
        int n1 = 7;

        Console.WriteLine("Ackermann value for m = " + m
                          + " and n = " + n + " is -> "
                          + Ackermann(m, n));

        Console.WriteLine("Ackermann value for m = " + m1
                          + " and n = " + n1 + " is -> "
                          + Ackermann(m1, n1));
    }
}

// This code is contributed by phasing17.

Javascript

// JS program to implement the approach

// Bottom Up Approach
function Ackermann(m, n)
{
    // creating 2D LIST
    let cache = new Array(m + 1);
    for (var i = 0; i <= m; i++)
        cache[i] = new Array(n + 1).fill(0);

    for (var rows = 0 ; rows < m + 1 ; rows++){
        for (var cols = 0 ; cols < n + 1 ; cols++){

            // base case A ( 0, n ) = n + 1
            if (rows == 0){
                cache[rows][cols] = cols + 1;
            }

            // base case A ( m, 0 ) =
            // A ( m-1, 1) [Computed already]
            else if (cols == 0)
            {
                cache[rows][cols] = cache[rows-1][1];
            }
            else
            {

                // if rows and cols > 0
                // then applying A ( m, n ) =
                // A ( m-1, A ( m, n-1 ) )
                let r = rows - 1;
                let c = cache[rows][cols-1];
                let ans;

                // applying equation (2)
                // here A ( 0, n ) = n + 1
                if (r == 0){
                    ans = c + 1;
                }else if (c <= n){
                    // using stored value in cache
                    ans = cache[rows-1][cache[rows][cols-1]];
                }else{
                    // Using the Derived Formula
                    // to compute mystery values in O(1) time
                    ans = (c-n)*(r) + cache[r][n];
                }

                cache[rows][cols] = ans;
            }
        }
    }
    return cache[m][n];
}


// Driver code

// very small values
let m = 2;
let n = 2;

// a bit higher value
let m1 = 5;
let n1 = 7;


console.log("Ackermann value for m = " + m +
      " and n = " + n + " is -> " +
      Ackermann(m, n));

console.log("Ackermann value for m = " +
      m1 + " and n = " + n1 + " is -> " +
      Ackermann(m1, n1));



// This code is contributed by phasing17.

Output

Ackermann value for m = 2 and n = 2 is -> 7
Ackermann value for m = 5 and n = 7 is -> 6141004759

Time complexity: O( M * N ) 
Auxiliary Space: O( M * N )