Given an array **arr**, the task is to remove minimum number of elements such that after their removal, **max(arr) <= 2 * min(arr)**.

**Examples:**

Input:arr[] = {4, 5, 3, 8, 3}Output:1

Remove 8 from the array.

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

Remove 1 from the array.

**Approach:** Let us fix each value as the minimum value say **x** and find number of terms that are in range **[x, 2*x]**. This can be done using prefix-sums, we can use map (implements self balancing BST) instead of array as the values can be large. The remaining terms which are not in range **[x, 2*x]** will have to be removed. So, across all values of **x**, we choose the one which maximises the number of terms in range **[x, 2*x]**.

Below is the implementation of the above approach:

## C++

`// C++ implementation of the approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` ` ` `// Function to return the minimum removals from ` `// arr such that max(arr) <= 2 * min(arr)` `int` `minimumRemovals(` `int` `n, ` `int` `a[])` `{` ` ` `// Count occurrence of each element` ` ` `map<` `int` `, ` `int` `> ct;` ` ` `for` `(` `int` `i = 0; i < n; i++)` ` ` `ct[a[i]]++;` ` ` ` ` `// Take prefix sum` ` ` `int` `sm = 0;` ` ` `for` `(` `auto` `mn : ct) {` ` ` `sm += mn.second;` ` ` `ct[mn.first] = sm;` ` ` `}` ` ` ` ` `int` `mx = 0, prev = 0;` ` ` `for` `(` `auto` `mn : ct) {` ` ` ` ` `// Chosen minimum` ` ` `int` `x = mn.first;` ` ` `int` `y = 2 * x;` ` ` `auto` `itr = ct.upper_bound(y);` ` ` `itr--;` ` ` ` ` `// Number of elements that are in` ` ` `// range [x, 2x]` ` ` `int` `cr = (itr->second) - prev;` ` ` `mx = max(mx, cr);` ` ` `prev = mn.second;` ` ` `}` ` ` ` ` `// Minimum elements to be removed` ` ` `return` `n - mx;` `}` ` ` `// Driver Program to test above function` `int` `main()` `{` ` ` `int` `arr[] = { 4, 5, 3, 8, 3 };` ` ` `int` `n = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]);` ` ` `cout << minimumRemovals(n, arr);` ` ` `return` `0;` `}` |

## Python3

`# Python3 implementation of the approach` `from` `bisect ` `import` `bisect_left as upper_bound` ` ` `# Function to return the minimum removals from` `# arr such that max(arr) <= 2 * min(arr)` `def` `minimumRemovals(n, a):` ` ` ` ` `# Count occurrence of each element` ` ` `ct ` `=` `dict` `()` ` ` `for` `i ` `in` `a:` ` ` `ct[i] ` `=` `ct.get(i, ` `0` `) ` `+` `1` ` ` ` ` `# Take prefix sum` ` ` `sm ` `=` `0` ` ` `for` `mn ` `in` `ct:` ` ` `sm ` `+` `=` `ct[mn]` ` ` `ct[mn] ` `=` `sm` ` ` ` ` `mx ` `=` `0` ` ` `prev ` `=` `0` `;` ` ` `for` `mn ` `in` `ct:` ` ` ` ` `# Chosen minimum` ` ` `x ` `=` `mn` ` ` `y ` `=` `2` `*` `x` ` ` `itr ` `=` `upper_bound(` `list` `(ct), y)` ` ` ` ` `# Number of elements that are in` ` ` `# range [x, 2x]` ` ` `cr ` `=` `ct[itr] ` `-` `prev` ` ` `mx ` `=` `max` `(mx, cr)` ` ` `prev ` `=` `ct[mn]` ` ` ` ` `# Minimum elements to be removed` ` ` `return` `n ` `-` `mx` ` ` `# Driver Code` `arr ` `=` `[` `4` `, ` `5` `, ` `3` `, ` `8` `, ` `3` `]` `n ` `=` `len` `(arr)` `print` `(minimumRemovals(n, arr))` ` ` `# This code is contributed by Mohit Kumar` |

**Output:**

1