Minimum removals from array to make max - min <= K

Given N integers and K, find the minimum number of elements that should be removed such that A_max-A_min<=K. After removal of elements, A_max and A_min is considered among the remaining elements.
Examples:

Input : a[] = {1, 3, 4, 9, 10, 11, 12, 17, 20} 
          k = 4 
Output : 5 
Explanation: Remove 1, 3, 4 from beginning 
and 17, 20 from the end.

Input : a[] = {1, 5, 6, 2, 8}  K=2
Output : 3
Explanation: There are multiple ways to 
remove elements in this case.
One among them is to remove 5, 6, 8.
The other is to remove 1, 2, 5

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: Sort the given elements. Using greedy approach, the best way is to remove the minimum element or the maximum element and then check if A_max-A_min <= K. There are various combinations of removals that have to be considered. We write a recurrence relation to try every possible combination. There will be two possible ways of removal, either we remove the minimum or we remove the maximum. Let(i…j) be the index of elements left after removal of elements. Initially, we start with i=0 and j=n-1 and the number of elements removed is 0 at the beginning. We only remove an element if a[j]-a[i]>k, the two possible ways of removal are (i+1…j) or (i…j-1). The minimum of the two is considered.

Let DP_{i, j} be the number of elements that need to be removed so that after removal a[j]-a[i]<=k.

Recurrence relation for sorted array:

DP_{i, j} = 1+ (min(countRemovals(i+1, j), countRemovals(i, j-1))

Below is the implementation of the above idea:

C++

// CPP program to find minimum removals 
// to make max-min <= K 
#include <bits/stdc++.h> 
using namespace std; 
  
#define MAX 100 
int dp[MAX][MAX]; 
  
// function to check all possible combinations 
// of removal and return the minimum one 
int countRemovals(int a[], int i, int j, int k) 
{ 
    // base case when all elements are removed 
    if (i >= j) 
        return 0; 
  
    // if condition is satisfied, no more 
    // removals are required 
    else if ((a[j] - a[i]) <= k) 
        return 0; 
  
    // if the state has already been visited 
    else if (dp[i][j] != -1) 
        return dp[i][j]; 
  
    // when Amax-Amin>d 
    else if ((a[j] - a[i]) > k) { 
  
        // minimum is taken of the removal 
        // of minimum element or removal 
        // of the maximum element 
        dp[i][j] = 1 + min(countRemovals(a, i + 1, j, k), 
                           countRemovals(a, i, j - 1, k)); 
    } 
    return dp[i][j]; 
} 
  
// To sort the array and return the answer 
int removals(int a[], int n, int k) 
{ 
    // sort the array 
    sort(a, a + n); 
  
    // fill all stated with -1 
    // when only one element 
    memset(dp, -1, sizeof(dp)); 
    if (n == 1) 
        return 0; 
    else
        return countRemovals(a, 0, n - 1, k); 
} 
  
// Driver Code 
int main() 
{ 
    int a[] = { 1, 3, 4, 9, 10, 11, 12, 17, 20 }; 
    int n = sizeof(a) / sizeof(a[0]); 
    int k = 4; 
    cout << removals(a, n, k); 
    return 0; 
} 

Java

// Java program to find minimum removals 
// to make max-min <= K 
import java.util.Arrays; 
  
class GFG 
{ 
    static int MAX=100; 
    static int dp[][]=new int[MAX][MAX]; 
      
    // function to check all possible combinations 
    // of removal and return the minimum one 
    static int countRemovals(int a[], int i, int j, int k) 
    { 
        // base case when all elements are removed 
        if (i >= j) 
            return 0; 
      
        // if condition is satisfied, no more 
        // removals are required 
        else if ((a[j] - a[i]) <= k) 
            return 0; 
      
        // if the state has already been visited 
        else if (dp[i][j] != -1) 
            return dp[i][j]; 
      
        // when Amax-Amin>d 
        else if ((a[j] - a[i]) > k) { 
      
            // minimum is taken of the removal 
            // of minimum element or removal 
            // of the maximum element 
            dp[i][j] = 1 + Math.min(countRemovals(a, i + 1, j, k), 
                                    countRemovals(a, i, j - 1, k)); 
        } 
        return dp[i][j]; 
    } 
      
    // To sort the array and return the answer 
    static int removals(int a[], int n, int k) 
    { 
        // sort the array 
        Arrays.sort(a); 
      
        // fill all stated with -1 
        // when only one element 
        for(int[] rows:dp) 
        Arrays.fill(rows,-1); 
        if (n == 1) 
            return 0; 
        else
            return countRemovals(a, 0, n - 1, k); 
    } 
      
    // Driver code 
    public static void main (String[] args)  
    { 
        int a[] = { 1, 3, 4, 9, 10, 11, 12, 17, 20 }; 
        int n = a.length; 
        int k = 4; 
        System.out.print(removals(a, n, k)); 
    } 
} 
  
// This code is contributed by Anant Agarwal. 

Python3

# Python program to find  
# minimum removals to 
# make max-min <= K 
MAX = 100
dp = [[0 for i in range(MAX)]  
         for i in range(MAX)] 
for i in range(0, MAX) : 
    for j in range(0, MAX) : 
        dp[i][j] = -1
  
# function to check all  
# possible combinations 
# of removal and return 
# the minimum one 
def countRemovals(a, i, j, k) : 
  
    global dp 
      
    # base case when all  
    # elements are removed 
    if (i >= j) : 
        return 0
  
    # if condition is satisfied,  
    # no more removals are required 
    elif ((a[j] - a[i]) <= k) : 
        return 0
  
    # if the state has  
    # already been visited 
    elif (dp[i][j] != -1) : 
        return dp[i][j] 
  
    # when Amax-Amin>d 
    elif ((a[j] - a[i]) > k) : 
  
        # minimum is taken of  
        # the removal of minimum 
        # element or removal of  
        # the maximum element 
        dp[i][j] = 1 + min(countRemovals(a, i + 1,  
                                         j, k), 
                           countRemovals(a, i,  
                                         j - 1, k)) 
    return dp[i][j] 
  
# To sort the array  
# and return the answer 
def removals(a, n, k) : 
  
    # sort the array 
    a.sort() 
  
    # fill all stated  
    # with -1 when only 
    # one element 
    if (n == 1) : 
        return 0
    else : 
        return countRemovals(a, 0,  
                             n - 1, k) 
  
# Driver Code 
a = [1, 3, 4, 9, 10,  
     11, 12, 17, 20] 
n = len(a) 
k = 4
print (removals(a, n, k)) 
  
# This code is contributed by  
# Manish Shaw(manishshaw1) 

C#

// C# program to find minimum  
// removals to make max-min <= K 
using System; 
  
class GFG 
{ 
    static int MAX = 100; 
    static int [,]dp = new int[MAX, MAX]; 
      
    // function to check all  
    // possible combinations 
    // of removal and return 
    // the minimum one 
    static int countRemovals(int []a, int i, 
                             int j, int k) 
    { 
        // base case when all 
        // elements are removed 
        if (i >= j) 
            return 0; 
      
        // if condition is satisfied,  
        // no more removals are required 
        else if ((a[j] - a[i]) <= k) 
            return 0; 
      
        // if the state has 
        // already been visited 
        else if (dp[i, j] != -1) 
            return dp[i, j]; 
      
        // when Amax-Amin>d 
        else if ((a[j] - a[i]) > k) 
        { 
      
            // minimum is taken of the  
            // removal of minimum element  
            // or removal of the maximum  
            // element 
            dp[i, j] = 1 + Math.Min(countRemovals(a, i + 1,  
                                                  j, k), 
                                    countRemovals(a, i,  
                                                  j - 1, k)); 
        } 
        return dp[i, j]; 
    } 
      
    // To sort the array and 
    // return the answer 
    static int removals(int []a,  
                        int n, int k) 
    { 
        // sort the array 
        Array.Sort(a); 
      
        // fill all stated with -1 
        // when only one element 
        for(int i = 0; i < MAX; i++)  
        { 
            for(int j = 0; j < MAX; j++) 
                dp[i, j] = -1; 
        } 
        if (n == 1) 
            return 0; 
        else
            return countRemovals(a, 0,  
                                 n - 1, k); 
    } 
      
    // Driver code 
    static void Main()  
    { 
        int []a = new int[]{ 1, 3, 4, 9, 10,  
                             11, 12, 17, 20 }; 
        int n = a.Length; 
        int k = 4; 
        Console.Write(removals(a, n, k)); 
    } 
} 
  
// This code is contributed by  
// ManishShaw(manishshaw1) 

PHP

<?php 
// PHP program to find  
// minimum removals to 
// make max-min <= K 
$MAX = 100; 
$dp = array(array()); 
for($i = 0; $i < $MAX; $i++) 
{ 
    for($j = 0; $j < $MAX; $j++) 
        $dp[$i][$j] = -1; 
} 
  
// function to check all  
// possible combinations 
// of removal and return 
// the minimum one 
function countRemovals($a, $i,  
                       $j, $k) 
{ 
    global $dp; 
      
    // base case when all  
    // elements are removed 
    if ($i >= $j) 
        return 0; 
  
    // if condition is satisfied,  
    // no more removals are required 
    else if (($a[$j] - $a[$i]) <= $k) 
        return 0; 
  
    // if the state has  
    // already been visited 
    else if ($dp[$i][$j] != -1) 
        return $dp[$i][$j]; 
  
    // when Amax-Amin>d 
    else if (($a[$j] - $a[$i]) > $k)  
    { 
  
        // minimum is taken of  
        // the removal of minimum 
        // element or removal of  
        // the maximum element 
        $dp[$i][$j] = 1 + min(countRemovals($a, $i + 1,  
                                                $j, $k), 
                              countRemovals($a, $i,  
                                            $j - 1, $k)); 
    } 
    return $dp[$i][$j]; 
} 
  
// To sort the array  
// and return the answer 
function removals($a, $n, $k) 
{ 
    // sort the array 
    sort($a); 
  
    // fill all stated with -1 
    // when only one element 
    if ($n == 1) 
        return 0; 
    else
        return countRemovals($a, 0,  
                             $n - 1, $k); 
} 
  
// Driver Code 
$a = array(1, 3, 4, 9, 10,  
           11, 12, 17, 20); 
$n = count($a); 
$k = 4; 
echo (removals($a, $n, $k)); 
  
// This code is contributed by  
// Manish Shaw(manishshaw1) 
?> 