In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. Refer this for more.
It's a function with two arguments each of which can be assigned any non-negative integer.
Ackermann function is defined as:
Ackermann algorithm:
Ackermann(m, n)
{next and goal are arrays indexed from 0 to m, initialized so that next[O]
through next[m] are 0, goal[O] through goal[m - l] are 1, and goal[m] is -1}
repeat
value <-- next[O] + 1
transferring <-- true
current <-- O
while transferring do begin
if next[current] = goal[current] then goal[current] <-- value
else transferring <-- false
next[current] <-- next[current]+l
current <-- current + 1
end while
until next[m] = n + 1
return value {the value of A(m, n)}
end Ackermann
Here's the explanation of the given Algorithm:
Let me explain the algorithm by taking the example A(1, 2) where m = 1 and n = 2
So according to the algorithm initially the value of next, goal, value and current are:
Though next[current] != goal[current], so else statement will execute and transferring become false.
So now, the value of next, goal, value and current are:
Similarly by tracing the algorithm until next[m] = 3 the value of next, goal, value and current are changing accordingly. Here's the explanation how the values are changing,
Finally returning the value e.g 4
Analysis of this algorithm:
- The time complexity of this algorithm is: O(mA(m, n)) to compute A(m, n)
- The space complexity of this algorithm is: O(m) to compute A(m, n)
Let's understand the definition by solving a problem!
Solve A(1, 2)?
Answer:
Given problem is A(1, 2)
Here m = 1, n = 2 e.g m > 0 and n > 0
Hence applying third condition of Ackermann function
A(1, 2) = A(0, A(1, 1)) ---------- (1)
Now, Let's find A(1, 1) by applying third condition of Ackermann function
A(1, 1) = A(0, A(1, 0)) ---------- (2)
Now, Let's find A(1, 0) by applying second condition of Ackermann function
A(1, 0) = A(0, 1) ---------- (3)
Now, Let's find A(0, 1) by applying first condition of Ackermann function
A(0, 1) = 1 + 1 = 2
Now put this value in equation 3
Hence A(1, 0) = 2
Now put this value in equation 2
A(1, 1) = A(0, 2) ---------- (4)
Now, Let's find A(0, 2) by applying first condition of Ackermann function
A(0, 2) = 2 + 1 = 3
Now put this value in equation 4
Hence A(1, 1) = 3
Now put this value in equation 1
A(1, 2) = A(0, 3) ---------- (5)
Now, Let's find A(0, 3) by applying first condition of Ackermann function
A(0, 3) = 3 + 1 = 4
Now put this value in equation 5
Hence A(1, 2) = 4
So, A (1, 2) = 4
Let's solve another two questions on this by yourself!
Question: Solve A(2, 1)?
Answer: 5
Question: Solve A(2, 2)?
Answer: 7
Here is the simplest c and python recursion function code for generating Ackermann function
C++
// C++ program to illustrate Ackermann function
#include <iostream>
using namespace std;
int ack(int m, int n)
{
if (m == 0){
return n + 1;
}
else if((m > 0) && (n == 0)){
return ack(m - 1, 1);
}
else if((m > 0) && (n > 0)){
return ack(m - 1, ack(m, n - 1));
}
}
// Driver code
int main()
{
int A;
A = ack(1, 2);
cout << A << endl;
return 0;
}
// This code is contributed by SHUBHAMSINGH10
// C program to illustrate Ackermann function
#include <stdio.h>
int ack(int m, int n)
{
if (m == 0){
return n+1;
}
else if((m > 0) && (n == 0)){
return ack(m-1, 1);
}
else if((m > 0) && (n > 0)){
return ack(m-1, ack(m, n-1));
}
}
int main(){
int A;
A = ack(1, 2);
printf("%d", A);
return 0;
}
// This code is contributed by Amiya Rout
// Java program to illustrate Ackermann function
class GFG
{
static int ack(int m, int n)
{
if (m == 0)
{
return n + 1;
}
else if((m > 0) && (n == 0))
{
return ack(m - 1, 1);
}
else if((m > 0) && (n > 0))
{
return ack(m - 1, ack(m, n - 1));
}else
return n + 1;
}
// Driver code
public static void main(String args[])
{
System.out.println(ack(1, 2));
}
}
// This code is contributed by AnkitRai01
# Python program to illustrate Ackermann function
def A(m, n, s ="% s"):
print(s % ("A(% d, % d)" % (m, n)))
if m == 0:
return n + 1
if n == 0:
return A(m - 1, 1, s)
n2 = A(m, n - 1, s % ("A(% d, %% s)" % (m - 1)))
return A(m - 1, n2, s)
print(A(1, 2))
# This code is contributed by Amiya Rout
// C# program to illustrate Ackermann function
using System;
class GFG
{
static int ack(int m, int n)
{
if (m == 0)
{
return n + 1;
}
else if((m > 0) && (n == 0))
{
return ack(m - 1, 1);
}
else if((m > 0) && (n > 0))
{
return ack(m - 1, ack(m, n - 1));
}else
return n + 1;
}
// Driver code
public static void Main()
{
Console.WriteLine(ack(1, 2));
}
}
// This code is contributed by mohit kumar 29
<script>
// Javascript program to illustrate Ackermann function
function ack(m,n)
{
if (m == 0)
{
return n + 1;
}
else if((m > 0) && (n == 0))
{
return ack(m - 1, 1);
}
else if((m > 0) && (n > 0))
{
return ack(m - 1, ack(m, n - 1));
}else
return n + 1;
}
// Driver code
document.write(ack(1, 2));
// This code is contributed by unknown2108
</script>
If you still wish to visualize how this result is arrived at, you can take a look at this page, which animates the calculation of every recursion step.
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem