# Find minimum possible size of array with given rules for removing elements

Given an array of numbers and a constant k, minimize size of array with following rules for removing elements.

• Exactly three elements can be removed at one go.
• The removed three elements must be adjacent in array, i.e., arr[i], arr[i+1], arr[i+2]. And the second element must be k greater than first and third element must be k greater than second, i.e., arr[i+1] – arr[i] = k and arr[i+2]-arr[i+1] = k.

Example:

```Input: arr[] = {2, 3, 4, 5, 6, 4}, k = 1
Output: 0
We can actually remove all elements.
First remove 4, 5, 6 => We get {2, 3, 4}
Now remove 2, 3, 4   => We get empty array {}

Input:  arr[] = {2, 3, 4, 7, 6, 4}, k = 1
Output: 3
We can only remove 2 3 4
```

We strongly recommend you to minimize your browser and try this yourself first.
For every element arr[i] there are two possibilities
1) Either the element is not removed.
2) OR element is removed (if it follows rules of removal). When an element is removed, there are again two possibilities.
…..a) It may be removed directly, i.e., initial arr[i+1] is arr[i]+k and arr[i+2] is arr[i] + 2*k.
…..b) There exist x and y such that arr[x] – arr[i] = k, arr[y] – arr[x] = k, and subarrays “arr[i+1…x-1]” & “arr[x+1…y-1]” can be completely removed.

Below is recursive algorithm based on above idea.

```// Returns size of minimum possible size of arr[low..high]
// after removing elements according to given rules
findMinSize(arr[], low, high, k)

// If there are less than 3 elements in arr[low..high]
1) If high-low+1 < 3, return high-low+1

// Consider the case when 'arr[low]' is not considered as
// part of any triplet to be removed.  Initialize result
// using this case
2) result = 1 + findMinSize(arr, low+1, high)

// Case when 'arr[low]' is part of some triplet and removed
// Try all possible triplets that have arr[low]
3) For all i from low+1 to high
For all j from i+1 to high
Update result if all of the following conditions are met
a) arr[i] - arr[low] = k
b) arr[j] - arr[i]  = k
c) findMinSize(arr, low+1, i-1, k) returns 0
d) findMinSize(arr, i+1, j-1, k) also returns 0
e) Result calculated for this triplet (low, i, j)
is smaller than existing result.

4) Return result
```

The time complexity of above solution is exponential. If we draw the complete recursion tree, we can observer that many subproblems are solved again and again. Since same suproblems are called again, this problem has Overlapping Subprolems property. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array dp[][] to store results of the subproblems. Below is Dynamic Programming based solution

Below is C++ implementation of above idea. The implementation is memoization based, i.e., it is recursive and uses a lookup table dp[][] to check if a subproblem is already solved or not.

```// C++ program to find size of minimum possible array after
// removing elements according to given rules
#include <bits/stdc++.h>
using namespace std;
#define MAX 1000

// dp[i][j] denotes the minimum number of elements left in
// the subarray arr[i..j].
int dp[MAX][MAX];

int minSizeRec(int arr[], int low, int high, int k)
{
if (dp[low][high] != -1)
return dp[low][high];

// If size of array is less than 3
if ( (high-low + 1) < 3)
return high-low +1;

// Initialize result as the case when first element is
// separated (not removed using given rules)
int res = 1 + minSizeRec(arr, low+1, high, k);

// Now consider all cases when first element forms a triplet
// and removed. Check for all possible triplets (low, i, j)
for (int i = low+1; i<=high-1; i++)
{
for (int j = i+1; j <= high; j++ )
{
// Check if this triplet follows the given rules of
// removal. And elements between 'low' and 'i' , and
//  between 'i' and 'j' can be recursively removed.
if (arr[i] == (arr[low] + k) &&
arr[j] == (arr[low] + 2*k) &&
minSizeRec(arr, low+1, i-1, k) == 0 &&
minSizeRec(arr, i+1, j-1, k) == 0)
{
res = min(res, minSizeRec(arr, j+1, high, k));
}
}
}

// Insert value in table and return result
return (dp[low][high] = res);
}

// This function mainlu initializes dp table and calls
// recursive function minSizeRec
int minSize(int arr[], int n, int k)
{
memset(dp, -1, sizeof(dp));
return minSizeRec(arr, 0, n-1, k);
}

// Driver prrogram to test above function
int main()
{
int arr[] = {2, 3, 4, 5, 6, 4};
int n = sizeof(arr)/sizeof(arr[0]);
int k = 1;
cout << minSize(arr, n, k) << endl;
return 0;
}
```

Output:

`0`

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