Given an array **A[]** consisting of **N** integers and an integer **K**, the task is to count the minimum number of distinct elements present in a subsequence of length **K **of the given array, **A**.

**Examples:**

Input:A = {3, 1, 3, 2, 3, 4, 5, 4}, K = 4Output:2Explanation:The subsequence of length 4 containing minimum number of distinct elements is {3, 3, 3, 4}, consisting of 2 distinct elements, i.e. {3, 4}.

Input:A = {3, 1, 3, 2, 3, 4, 5, 4}, K = 5Output:2Explanation:The subsequence of length 5 containing minimum number of distinct elements is {3, 3, 3, 4, 4}, consisting of 2 distinct elements, i.e. {3, 4}.

**Naive Approach: **The simplest approach is to generate all subsequences of length **K**** **and for each subsequence, find the number of distinct elements present in them. Finally, print the minimum number of distinct elements present.

**Time Complexity: **O(K * N^{K})**Auxiliary Space: **O(N)

**Efficient Approach:** The above approach can be optimized using **Hashing****.** Follow the steps below to solve the problem:

- Store the frequencies of all elements in the given array,
**A[]**in a HashMap, say**M**. - Traverse the hashmap,
**M**and push the frequencies in another array, say**V**. - Sort the array
**V**in decreasing order. - Initialize two variables,
**cnt**and**len**as**0,**to store the required result and the length of the subsequence thus formed. - Traverse the array
**V[]**using a variable, say**i**- If the value of
**len ≥ K,**then break out of the loop. - Otherwise, increment the value of
**len**by**V[i]**and**cnt**by**1**.

- If the value of
- After completing the above steps, print the value of
**cnt**as the result.

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to count the minimum number` `// of distinct elements present in any` `// subsequence of length K of the given array` `void` `findMinimumDistinct(` `int` `A[], ` `int` `N, ` `int` `K)` `{` ` ` `// Stores the frequency` ` ` `// of each array element` ` ` `unordered_map<` `int` `, ` `int` `> mp;` ` ` `// Traverse the array` ` ` `for` `(` `int` `i = 0; i < N; i++)` ` ` `// Update frequency` ` ` `// of array elements` ` ` `mp[A[i]]++;` ` ` `// Store the required result` ` ` `int` `count = 0;` ` ` `// Store the length of the` ` ` `// required subsequence` ` ` `int` `len = 0;` ` ` `// Store the frequencies` ` ` `// in decreasing order` ` ` `vector<` `int` `> counts;` ` ` `// Travere the map` ` ` `for` `(` `auto` `i : mp)` ` ` `// Push the frequencies` ` ` `// into the HashMap` ` ` `counts.push_back(i.second);` ` ` `// Sort the array in decreasing order` ` ` `sort(counts.begin(), counts.end(),` ` ` `greater<` `int` `>());` ` ` `// Add the elements into the subsequence` ` ` `// starting from one with highest frequency` ` ` `for` `(` `int` `i = 0; i < counts.size(); i++) {` ` ` `// If length of subsequence is >= k` ` ` `if` `(len >= K)` ` ` `break` `;` ` ` `len += counts[i];` ` ` `count++;` ` ` `}` ` ` `// Print the result` ` ` `cout << count;` `}` `// Driver Code` `int` `main()` `{` ` ` `int` `A[] = { 3, 1, 3, 2, 3, 4, 5, 4 };` ` ` `int` `K = 4;` ` ` `// Store the size of the array` ` ` `int` `N = ` `sizeof` `(A) / ` `sizeof` `(A[0]);` ` ` `// Function Call to count minimum` ` ` `// number of distinct elmeents` ` ` `// present in a K-length subsequence` ` ` `findMinimumDistinct(A, N, K);` ` ` `return` `0;` `}` |

## Python3

`# Python3 program for the above approach` `from` `collections ` `import` `Counter` `# Function to count the minimum number` `# of distinct elements present in any` `# subsequence of length K of the given array` `def` `findMinimumDistinct(A, N, K):` ` ` ` ` `# Stores the frequency` ` ` `# of each array element` ` ` `mp ` `=` `Counter(A)` ` ` ` ` `# Store the required result` ` ` `count ` `=` `0` ` ` ` ` `# Store the length of the` ` ` `# required subsequence` ` ` `length ` `=` `0` ` ` ` ` `# Store the frequencies` ` ` `# in decreasing order` ` ` `counts ` `=` `[]` ` ` ` ` `# Travere the map` ` ` `for` `i ` `in` `mp:` ` ` ` ` `# Push the frequencies` ` ` `# into the HashMap` ` ` `counts.append(mp[i])` ` ` ` ` `# Sort the array in decreasing order` ` ` `counts ` `=` `sorted` `(counts)` ` ` `counts.reverse()` ` ` ` ` `# Add the elements into the subsequence` ` ` `# starting from one with highest frequency` ` ` `for` `i ` `in` `range` `(` `len` `(counts)):` ` ` ` ` `# If length of subsequence is >= k` ` ` `if` `(length >` `=` `K):` ` ` `break` ` ` ` ` `length ` `+` `=` `counts[i]` ` ` `count ` `+` `=` `1` ` ` ` ` `# Print the result` ` ` `print` `(count)` `# Driver Code` `A ` `=` `[` `3` `, ` `1` `, ` `3` `, ` `2` `, ` `3` `, ` `4` `, ` `5` `, ` `4` `]` `K ` `=` `4` `# Store the size of the array` `N ` `=` `len` `(A)` `# Function Call to count minimum` `# number of distinct elmeents` `# present in a K-length subsequence` `findMinimumDistinct(A, N, K)` `# This code is contributed by sudhanshugupta2019a` |

**Output:**

2

**Time Complexity: **O(N*log(N))**Auxiliary Space: **O(N)

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