Count minimum number of "move-to-front" moves to sort an array

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Given an array of size n such that array elements are in range from 1 to n. The task is to count a number of move-to-front operations to arrange items as {1, 2, 3,... n}. The move-to-front operation is to pick any item and place it in first position.

This problem can also be seen as a stack of items with only move available is to pull an item from the stack and placing it on top of the stack. 

Examples : 

Input: arr[] = {3, 2, 1, 4}.
Output: 2
First, we pull out 2 and places it on top, 
so the array becomes (2, 3, 1, 4). After that, 
pull out 1 and becomes (1, 2, 3, 4).

Input:  arr[] = {5, 7, 4, 3, 2, 6, 1}
Output:  6
We pull elements in following order
7, 6, 5, 4, 3 and 2

Input: arr[] = {4, 3, 2, 1}.
Output: 3

The idea is to traverse array from end. We expect n at the end, so we initialize expectedItem as n. All the items which are between actual position of expectedItem and current position must be moved to front. So we calculate the number of items between current item and expected item. Once we find expectedItem, we look for next expectedItem by reducing expectedITem by one. 

Following is the algorithm for the minimum number of moves: 

1. Initialize expected number at current position as n 
2. Start from the last element of array.
     a) If the current item is same as expected item, 
           decrease expected item by 1.
3. Return expected item.

Below is the implementation of this approach. 

// C++ program to find minimum number of move-to-front
// moves to arrange items in sorted order.
#include <bits/stdc++.h>
using namespace std;

// Calculate minimum number of moves to arrange array
// in increasing order.
int minMoves(int arr[], int n)
{
    // Since we traverse array from end, expected item
    // is initially  n
    int expectedItem = n;

    // Traverse array from end
    for (int i=n-1; i >= 0; i--)
    {
        // If current item is at its correct position,
        // decrement the expectedItem (which also means
        // decrement in minimum number of moves)
        if (arr[i] == expectedItem)
            expectedItem--;
    }

    return expectedItem;
}

// Driver Program
int main()
{
    int arr[] = {4, 3, 2, 1};
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << minMoves(arr, n);
    return 0;
}

// java program to find minimum 
// number of move-to-front moves 
// to arrange items in sorted order.
import java.io.*;

class GFG 
{
    // Calculate minimum number of moves 
    // to arrange array in increasing order.
    static int minMoves(int arr[], int n)
    {
        // Since we traverse array from end, 
        // expected item is initially n
        int expectedItem = n;
    
        // Traverse array from end
        for (int i = n - 1; i >= 0; i--)
        {
            // If current item is at its correct position,
            // decrement the expectedItem (which also means
            // decrement in minimum number of moves)
            if (arr[i] == expectedItem)
                expectedItem--;
        }
    
        return expectedItem;
    }
    
    // Driver Program
    public static void main (String[] args) 
    {
        int arr[] = {4, 3, 2, 1};
        int n = arr.length;
        System.out.println( minMoves(arr, n));

    }
}

// This code is contributed by vt_m

# Python 3 program to find minimum
# number of move-to-front moves
# to arrange items in sorted order.

# Calculate minimum number of moves 
# to arrange array in increasing order.
def minMoves(arr, n):
    
    # Since we traverse array from end, 
    # expected item is initially n
    expectedItem = n

    # Traverse array from end
    for i in range(n - 1, -1, -1):
        
        # If current item is at its
        # correct position, decrement
        # the expectedItem (which also
        # means decrement in minimum
        # number of moves)
        if (arr[i] == expectedItem):
            expectedItem -= 1
    return expectedItem
    
# Driver Code
arr = [4, 3, 2, 1]
n = len(arr)
print(minMoves(arr, n))

# This code is contributed 29AjayKumar

// C# program to find minimum 
// number of move-to-front moves 
// to arrange items in sorted order.
using System;

class GFG {

    // Calculate minimum number of moves 
    // to arrange array in increasing order.
    static int minMoves(int []arr, int n)
    {
        // Since we traverse array from end, 
        // expected item is initially n
        int expectedItem = n;
    
        // Traverse array from end
        for (int i = n - 1; i >= 0; i--)
        {
            // If current item is at its
            // correct position, decrement 
            // the expectedItem (which also
            // means decrement in minimum
            // number of moves)
            if (arr[i] == expectedItem)
                expectedItem--;
        }
    
        return expectedItem;
    }
    
    // Driver Program
    public static void Main () 
    {
        int []arr = {4, 3, 2, 1};
        int n = arr.Length;
        Console.Write( minMoves(arr, n));

    }
}

// This code is contributed by nitin mittal.

<?php
// PHP program to find minimum 
// number of move-to-front moves
// to arrange items in sorted order.

// Calculate minimum number of
// moves to arrange array
// in increasing order.
function minMoves($arr, $n)
{
    // Since we traverse array 
    // from end, expected item
    // is initially n
    $expectedItem = $n;

    // Traverse array from end
    for ($i = $n - 1; $i >= 0; $i--)
    {
        // If current item is at its 
        // correct position, decrement 
        // the expectedItem (which also 
        // means decrement in minimum 
        // number of moves)
        if ($arr[$i] == $expectedItem)
            $expectedItem--;
    }

    return $expectedItem;
}

// Driver Code
$arr = array(4, 3, 2, 1);
$n = count($arr);
echo minMoves($arr, $n);

// This code is contributed by anuj_67.
?>

<script>

    // JavaScript program to find minimum 
    // number of move-to-front moves 
    // to arrange items in sorted order.
    
    // Calculate minimum number of moves 
    // to arrange array in increasing order.
    function minMoves(arr, n)
    {
        // Since we traverse array from end, 
        // expected item is initially n
        let expectedItem = n;
      
        // Traverse array from end
        for (let i = n - 1; i >= 0; i--)
        {
            // If current item is at its
            // correct position, decrement 
            // the expectedItem (which also
            // means decrement in minimum
            // number of moves)
            if (arr[i] == expectedItem)
                expectedItem--;
        }
      
        return expectedItem;
    }
    
    let arr = [4, 3, 2, 1];
    let n = arr.length;
    document.write( minMoves(arr, n));
    
</script>

3

Time Complexity : O(n)

Auxiliary Space: O(1)