# Count minimum number of moves to front or end to sort an array

Given an array arr[] of size N, the task is to find the minimum moves to the beginning or end of the array required to make the array sorted in non-decreasing order.

Examples:

Input: arr[] = {4, 7, 2, 3, 9}
Output:
Explanation:
Perform the following operations:
Step 1: Move the element 3 to the start of the array. Now, arr[] modifies to {3, 4, 7, 2, 9}.
Step 2: Move the element 2 to the start of the array. Now, arr[] modifies to {2, 3, 4, 7, 9}.
Now, the resultant array is sorted.
Therefore, the minimum moves required is 2.

Input: arr[] = {1, 4, 5, 7, 12}
Output:
Explanation:
The array is already sorted. Therefore, no moves required.

Naive Approach: The simplest approach is to check for every array element, how many moves are required to sort the given array arr[]. For each array element, if it is not at its sorted position, the following possibilities arise:

• Either move the current element to the front.
• Otherwise, move the current element to the end.

After performing the above operations, print the minimum number of operations required to make the array sorted. Below is the recurrence relation of the same:

• If the array arr[] is equal to the array brr[], then return 0.
• If arr[i] < brr[j], then count of operation will be:

1 + recursive_function(arr, brr, i + 1, j + 1)

• Otherwise, the count of operation can be calculated by taking the maximum of the following states:
1. recursive_function(arr, brr, i + 1, j)
2. recursive_function(arr, brr, i, j + 1)

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ` `using` `namespace` `std;`   `// Function that counts the minimum` `// moves required to covert arr[] to brr[]` `int` `minOperations(``int` `arr1[], ``int` `arr2[], ` `                  ``int` `i, ``int` `j,` `                  ``int` `n)` `{` `  ``// Base Case` `  ``int` `f = 0;` `  ``for` `(``int` `i = 0; i < n; i++) ` `  ``{` `    ``if` `(arr1[i] != arr2[i])` `      ``f = 1;` `    ``break``;` `  ``}` `  ``if` `(f == 0)` `    ``return` `0;`   `  ``if` `(i >= n || j >= n)` `    ``return` `0;`   `  ``// If arr[i] < arr[j]` `  ``if` `(arr1[i] < arr2[j])`   `    ``// Include the current element` `    ``return` `1 + minOperations(arr1, arr2, ` `                             ``i + 1, j + 1, n);`   `  ``// Otherwise, excluding the current element` `  ``return` `max(minOperations(arr1, arr2, ` `                           ``i, j + 1, n),` `             ``minOperations(arr1, arr2, ` `                           ``i + 1, j, n));` `}`   `// Function that counts the minimum` `// moves required to sort the array` `void` `minOperationsUtil(``int` `arr[], ``int` `n)` `{` `  ``int` `brr[n];`   `  ``for` `(``int` `i = 0; i < n; i++)` `    ``brr[i] = arr[i];`   `  ``sort(brr, brr + n);` `  ``int` `f = 0;`   `  ``// If both the arrays are equal` `  ``for` `(``int` `i = 0; i < n; i++) ` `  ``{` `    ``if` `(arr[i] != brr[i])`   `      ``// No moves required` `      ``f = 1;` `    ``break``;` `  ``}` `  `  `  ``// Otherwise` `  ``if` `(f == 1)` `    `  `    ``// Print minimum ` `    ``// operations required` `    ``cout << (minOperations(arr, brr, ` `                           ``0, 0, n));` `  ``else` `    ``cout << ``"0"``;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `arr[] = {4, 7, 2, 3, 9};` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``minOperationsUtil(arr, n);` `}`   `// This code is contributed by Chitranayal`

## Java

 `// Java program for the above approach ` `import` `java.util.*;` `import` `java.io.*;` `import` `java.lang.Math;`   `class` `GFG{`   `// Function that counts the minimum ` `// moves required to covert arr[] to brr[] ` `static` `int` `minOperations(``int` `arr1[], ``int` `arr2[],` `                         ``int` `i, ``int` `j)` `{` `    `  `    ``// Base Case ` `    ``if` `(arr1.equals(arr2))` `        ``return` `0``;` `          `  `    ``if` `(i >= arr1.length || j >= arr2.length)` `        ``return` `0``;` `      `  `    ``// If arr[i] < arr[j] ` `    ``if` `(arr1[i] < arr2[j]) ` `    `  `        ``// Include the current element ` `        ``return` `1` `+ minOperations(arr1, arr2, ` `                                 ``i + ``1``, j + ``1``);` `         `  `    ``// Otherwise, excluding the current element ` `    ``return` `Math.max(minOperations(arr1, arr2, ` `                                  ``i, j + ``1``), ` `                    ``minOperations(arr1, arr2,` `                                  ``i + ``1``, j)); ` `}`   `// Function that counts the minimum ` `// moves required to sort the array ` `static` `void` `minOperationsUtil(``int``[] arr)` `{ ` `    ``int` `brr[] = ``new` `int``[arr.length];` `    `  `    ``for``(``int` `i = ``0``; i < arr.length; i++)` `        ``brr[i] = arr[i];` `        `  `    ``Arrays.sort(brr);` `      `  `    ``// If both the arrays are equal ` `    ``if` `(arr.equals(brr))` `    `  `        ``// No moves required ` `        ``System.out.print(``"0"``);` `        `  `    ``// Otherwise ` `    ``else` `    `  `        ``// Print minimum operations required ` `        ``System.out.println(minOperations(arr, brr,` `                                         ``0``, ``0``));` `}`   `// Driver code` `public` `static` `void` `main(``final` `String[] args)` `{` `    ``int` `arr[] = { ``4``, ``7``, ``2``, ``3``, ``9` `};` `    `  `    ``minOperationsUtil(arr); ` `}` `}`   `// This code is contributed by bikram2001jha`

## Python3

 `# Python3 program for the above approach`   `# Function that counts the minimum` `# moves required to covert arr[] to brr[]` `def` `minOperations(arr1, arr2, i, j):` `    `  `    ``# Base Case` `    ``if` `arr1 ``=``=` `arr2:` `        ``return` `0` `        `  `    ``if` `i >``=` `len``(arr1) ``or` `j >``=` `len``(arr2):` `        ``return` `0` `    `  `    ``# If arr[i] < arr[j]` `    ``if` `arr1[i] < arr2[j]:` `        `  `        ``# Include the current element` `        ``return` `1` `\` `        ``+` `minOperations(arr1, arr2, i ``+` `1``, j ``+` `1``)` `        `  `    ``# Otherwise, excluding the current element` `    ``return` `max``(minOperations(arr1, arr2, i, j ``+` `1``),` `               ``minOperations(arr1, arr2, i ``+` `1``, j))` `    `  `# Function that counts the minimum` `# moves required to sort the array` `def` `minOperationsUtil(arr):` `    `  `    ``brr ``=` `sorted``(arr);` `    `  `    ``# If both the arrays are equal` `    ``if``(arr ``=``=` `brr):` `        `  `        ``# No moves required` `        ``print``(``"0"``)`   `    ``# Otherwise` `    ``else``:` `        `  `        ``# Print minimum operations required` `        ``print``(minOperations(arr, brr, ``0``, ``0``))`   `# Driver Code`   `arr ``=` `[``4``, ``7``, ``2``, ``3``, ``9``]`   `minOperationsUtil(arr)`

## C#

 `// C# program for the above approach` `using` `System;`   `class` `GFG{` `    `  `// Function that counts the minimum ` `// moves required to covert arr[] to brr[] ` `static` `int` `minOperations(``int``[] arr1, ``int``[] arr2,` `                         ``int` `i, ``int` `j)` `{` `    `  `    ``// Base Case ` `    ``if` `(arr1.Equals(arr2))` `        ``return` `0;` `           `  `    ``if` `(i >= arr1.Length || ` `        ``j >= arr2.Length)` `        ``return` `0;` `       `  `    ``// If arr[i] < arr[j] ` `    ``if` `(arr1[i] < arr2[j]) ` `     `  `        ``// Include the current element ` `        ``return` `1 + minOperations(arr1, arr2, ` `                                 ``i + 1, j + 1);` `          `  `    ``// Otherwise, excluding the current element ` `    ``return` `Math.Max(minOperations(arr1, arr2, ` `                                  ``i, j + 1), ` `                    ``minOperations(arr1, arr2,` `                                  ``i + 1, j)); ` `}`   `// Function that counts the minimum ` `// moves required to sort the array ` `static` `void` `minOperationsUtil(``int``[] arr)` `{ ` `    ``int``[] brr = ``new` `int``[arr.Length];` `     `  `    ``for``(``int` `i = 0; i < arr.Length; i++)` `        ``brr[i] = arr[i];` `         `  `    ``Array.Sort(brr);` `       `  `    ``// If both the arrays are equal ` `    ``if` `(arr.Equals(brr))` `     `  `        ``// No moves required ` `        ``Console.Write(``"0"``);` `         `  `    ``// Otherwise ` `    ``else` `     `  `        ``// Print minimum operations required ` `        ``Console.WriteLine(minOperations(arr, brr,` `                                         ``0, 0));` `}`   `// Driver code` `static` `void` `Main()` `{` `    ``int``[] arr = { 4, 7, 2, 3, 9 };` `     `  `    ``minOperationsUtil(arr); ` `}` `}`   `// This code is contributed by divyeshrabadiya07`

Output:

```2

```

Time Complexity: O(2N
Auxiliary Space: O(N)

Efficient Approach: The above approach has many overlapping subproblems. Therefore, the above approach can be optimized using Dynamic programming. Follow the steps below to solve the problem:

• Maintain a 2D array table[][] to store the computed results.
• Apply recursion to solve the problem using the results of smaller subproblems.
• If arr1[i] < arr2[j], then return 1 + minOperations(arr1, arr2, i + 1, j – 1, table)
• Otherwise, either move the i-th element of the array to the end or the j-th element of the array to the front. Therefore, the recurrence relation is:

table[i][j] = max(minOperations(arr1, arr2, i, j + 1, table), minOperations(arr1, arr2, i + 1, j, table))

• Finally, print the value stored in table[0][N – 1].

Below is the implementation of the above approach:

## Python3

 `# Python3 program for the above approach `   `# Function to find the minimum number ` `# of moves required to sort the array ` `def` `minOperations(arr1, arr2, i, j, table): ` `    `  `    ``key ``=` `(i, j) `   `    ``# Base Cases ` `    ``if` `arr1 ``=``=` `arr2: ` `        ``return` `0` `    ``if` `i >``=` `len``(arr1) ``or` `j >``=` `len``(arr2): ` `        ``return` `0`   `    ``# If the result is already stored ` `    ``if` `key ``in` `table: ` `        ``return` `table[key] `   `    ``# Compute the result ` `    ``if` `arr1[i] < arr2[j]: ` `        ``return` `1` `+` `minOperations(` `                   ``arr1, arr2, i ``+` `1``, ` `                   ``j ``+` `1``, table) `   `    ``# Store the result ` `    ``table[key] ``=` `max``( ` `        ``minOperations(arr1, arr2, i, ` `        ``j ``+` `1``, table), ` `        `  `        ``minOperations(arr1, arr2, i ``+` `1``, ` `        ``j, table)) `   `    ``# Return the result ` `    ``return` `table[key] `   `# Function to print the minimum ` `# moves required to sort the array ` `def` `minOperationsUtil(arr): `   `    ``brr ``=` `sorted``(arr) ` `    ``table ``=` `dict``() ` `    `  `    ``# If both the arrays are equal ` `    ``if` `brr ``=``=` `arr: ` `        `  `        ``# No moves required ` `        ``print``(``0``) `   `    ``# Otherwise ` `    ``else``: ` `        ``print``(``len``(brr) ``-` `              ``minOperations(brr, arr, ``0``, ``0``, table)) `   `# Driver Code `   `arr ``=` `[``4``, ``7``, ``2``, ``3``, ``9``] `   `minOperationsUtil(arr) `

Output:

```2

```

Time Complexity: O(N)

Auxiliary Space: O(N)

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