Minimize Steps required to obtain Sorted Order of an Array

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Given an array arr[] consisting of a permutation of integers [1, N], derived by rearranging the sorted order [1, N], the task is to find the minimum number of steps after which the sorted order [1, N] is repeated, by repeating the same process by which arr[] is obtained from the sorted sequence at each step.

Examples:

Input: arr[ ] = {3, 6, 5, 4, 1, 2}
Output: 6
Explanation:
Increasing Permutation: {1, 2, 3, 4, 5, 6}
Step 1 : arr[] = {3, 6, 5, 4, 1, 2} (Given array)
Step 2 : arr[] = {5, 2, 1, 4, 3, 6}
Step 3 : arr[] = {1, 6, 3, 4, 5, 2}
Step 4 : arr[] = {3, 2, 5, 4, 1, 6}
Step 5 : arr[] = {5, 6, 1, 4, 3, 2}
Step 6 : arr[] = {1, 2, 3, 4, 5, 6} (Increasing Permutation)
Therefore, the total number of steps required are 6.
Input: arr[ ] = [5, 1, 4, 3, 2]
Output: 6

Approach:
This problem can be solved simply by using the concept of Direct Addressing. Follow the steps given below to solve the problem:

Initialize an array dat[] for direct addressing.
Iterate over [1, N] and calculate the difference of the current index of every element from its index in the sorted sequence.
Calculate the LCM of the array dat[].
Now, print the obtained LCM as the minimum steps required to obtain the sorted order.

Below is the implementation of the above approach:

C++14

// C++ Program to implement 
// the above approach 
#include <bits/stdc++.h> 
using namespace std; 
 
// Function to find 
// GCD of two numbers 
int gcd(int a, int b) 
{ 
    if (b == 0) 
        return a; 
 
    return gcd(b, a % b); 
} 
 
// Function to calculate the 
// LCM of array elements 
int findlcm(int arr[], int n) 
{ 
    // Initialize result 
    int ans = 1; 
 
    for (int i = 1; i <= n; i++) 
        ans = (((arr[i] * ans)) 
            / (gcd(arr[i], ans))); 
 
    return ans; 
} 
 
// Function to find minimum steps 
// required to obtain sorted sequence 
void minimumSteps(int arr[], int n) 
{ 
 
    // Initialize dat[] array for 
    // Direct Address Table. 
    int i, dat[n + 1]; 
 
    for (i = 1; i <= n; i++) 
 
        dat[arr[i - 1]] = i; 
 
    int b[n + 1], j = 0, c; 
 
    // Calculating steps required 
    // for each element to reach 
    // its sorted position 
    for (i = 1; i <= n; i++) { 
        c = 1; 
        j = dat[i]; 
        while (j != i) { 
            c++; 
            j = dat[j]; 
        } 
        b[i] = c; 
    } 
 
    // Calculate LCM of the array 
    cout << findlcm(b, n); 
} 
 
// Driver Code 
int main() 
{ 
 
    int arr[] = { 5, 1, 4, 3, 2, 7, 6 }; 
 
    int N = sizeof(arr) / sizeof(arr[0]); 
 
    minimumSteps(arr, N); 
 
    return 0; 
} 

Java

// Java program to implement
// the above approach
class GFG{
     
// Function to find
// GCD of two numbers
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
 
    return gcd(b, a % b);
}
 
// Function to calculate the
// LCM of array elements
static int findlcm(int arr[], int n)
{
     
    // Initialize result
    int ans = 1;
 
    for(int i = 1; i <= n; i++)
        ans = (((arr[i] * ans)) / 
            (gcd(arr[i], ans)));
 
    return ans;
}
 
// Function to find minimum steps
// required to obtain sorted sequence
static void minimumSteps(int arr[], int n)
{
 
    // Initialize dat[] array for
    // Direct Address Table.
    int i;
    int dat[] = new int[n + 1];
 
    for(i = 1; i <= n; i++)
        dat[arr[i - 1]] = i;
 
    int b[] = new int[n + 1];
    int j = 0, c;
 
    // Calculating steps required
    // for each element to reach
    // its sorted position
    for(i = 1; i <= n; i++)
    {
        c = 1;
        j = dat[i];
         
        while (j != i) 
        {
            c++;
            j = dat[j];
        }
        b[i] = c;
    }
 
    // Calculate LCM of the array
    System.out.println(findlcm(b, n));
}
 
// Driver code    
public static void main(String[] args)
{
    int arr[] = { 5, 1, 4, 3, 2, 7, 6 };
 
    int N = arr.length;
 
    minimumSteps(arr, N);
}
}
 
// This code is contributed by rutvik_56

Python3

# Python3 program to implement 
# the above approach 
 
# Function to find 
# GCD of two numbers 
def gcd(a, b): 
 
    if(b == 0): 
        return a 
 
    return gcd(b, a % b) 
 
# Function to calculate the 
# LCM of array elements 
def findlcm(arr, n): 
 
    # Initialize result 
    ans = 1
 
    for i in range(1, n + 1): 
        ans = ((arr[i] * ans) //
            (gcd(arr[i], ans))) 
 
    return ans 
 
# Function to find minimum steps 
# required to obtain sorted sequence 
def minimumSteps(arr, n): 
 
    # Initialize dat[] array for 
    # Direct Address Table. 
    dat = [0] * (n + 1) 
 
    for i in range(1, n + 1): 
        dat[arr[i - 1]] = i 
 
    b = [0] * (n + 1) 
    j = 0
 
    # Calculating steps required 
    # for each element to reach 
    # its sorted position 
    for i in range(1, n + 1): 
        c = 1
        j = dat[i] 
        while(j != i): 
            c += 1
            j = dat[j] 
 
        b[i] = c 
 
    # Calculate LCM of the array 
    print(findlcm(b, n)) 
 
# Driver Code 
arr = [ 5, 1, 4, 3, 2, 7, 6 ] 
 
N = len(arr) 
 
minimumSteps(arr, N) 
 
# This code is contributed by Shivam Singh 

C#

// C# program to implement
// the above approach
using System;
 
class GFG{
     
// Function to find
// GCD of two numbers
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
 
    return gcd(b, a % b);
}
 
// Function to calculate the
// LCM of array elements
static int findlcm(int []arr, int n)
{
     
    // Initialize result
    int ans = 1;
 
    for(int i = 1; i <= n; i++)
        ans = (((arr[i] * ans)) / 
            (gcd(arr[i], ans)));
 
    return ans;
}
 
// Function to find minimum steps
// required to obtain sorted sequence
static void minimumSteps(int []arr, int n)
{
 
    // Initialize dat[] array for
    // Direct Address Table.
    int i;
    int []dat = new int[n + 1];
 
    for(i = 1; i <= n; i++)
        dat[arr[i - 1]] = i;
 
    int []b = new int[n + 1];
    int j = 0, c;
 
    // Calculating steps required
    // for each element to reach
    // its sorted position
    for(i = 1; i <= n; i++)
    {
        c = 1;
        j = dat[i];
         
        while (j != i) 
        {
            c++;
            j = dat[j];
        }
        b[i] = c;
    }
 
    // Calculate LCM of the array
    Console.WriteLine(findlcm(b, n));
}
 
// Driver code 
public static void Main(String[] args)
{
    int []arr = { 5, 1, 4, 3, 2, 7, 6 };
 
    int N = arr.Length;
 
    minimumSteps(arr, N);
}
}
 
// This code is contributed by gauravrajput1 

Javascript

<script>
// JavaScript program for the above approach
 
// Function to find
// GCD of two numbers
function gcd(a, b)
{
    if (b == 0)
        return a;
   
    return gcd(b, a % b);
}
   
// Function to calculate the
// LCM of array elements
function findlcm(arr, n)
{
       
    // Initialize result
    let ans = 1;
   
    for(let i = 1; i <= n; i++)
        ans = (((arr[i] * ans)) / 
            (gcd(arr[i], ans)));
   
    return ans;
}
   
// Function to find minimum steps
// required to obtain sorted sequence
function minimumSteps(arr, n)
{
   
    // Initialize dat[] array for
    // Direct Address Table.
    let i;
    let dat = Array.from({length: n+1}, (_, i) => 0);
   
    for(i = 1; i <= n; i++)
        dat[arr[i - 1]] = i;
   
    let b = Array.from({length: n+1}, (_, i) => 0);
    let j = 0, c;
   
    // Calculating steps required
    // for each element to reach
    // its sorted position
    for(i = 1; i <= n; i++)
    {
        c = 1;
        j = dat[i];
           
        while (j != i) 
        {
            c++;
            j = dat[j];
        }
        b[i] = c;
    }
   
    // Calculate LCM of the array
    document.write(findlcm(b, n));
}
     
// Driver Code    
     
    let arr = [ 5, 1, 4, 3, 2, 7, 6 ];
   
    let N = arr.length;
   
    minimumSteps(arr, N);
                   
</script>

Output:

Time Complexity: O(NlogN)
Auxiliary Space: O(N)

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