Number of subsequences of maximum length K containing no repeated elements

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Given an array arr[] of N elements and a positive integer K such that K ≤ N. The task is to find the number of subsequences of maximum length K i.e. subsequences of length 0, 1, 2, …, K – 1, K that have all distinct elements.

Examples:

Input: arr[] = {2, 2, 3, 3, 5}, K = 2
Output: 14
All the valid subsequences are {}, {2}, {2}, {3}, {3}, {5},
{2, 3}, {2, 3}, {2, 3}, {2, 3}, {2, 5}, {2, 5}, {3, 5} and {3, 5}.

Input: arr[] = {1, 2, 3, 4, 4}, K = 4
Output: 24

Approach:

Sort the array a[] if it is not already sorted and in a vector arr[] store the frequencies of each element of the original array. For example, if a[] = {2, 2, 3, 3, 5} then arr[] = {2, 2, 1} because 2 is present twice, 3 is present twice and 5 only once.
Say m is the length of the vector arr[]. So m will be the number of distinct elements. There can be subsequences of maximum length m without repetition. If m < k then there is no subsequence of length k. So, declare n = minimum(m, k).
Now apply dynamic programming. Create a 2-d array dp[n + 1][m + 1] such that dp[i][j] will store the number of subsequences of length i whose first element begins after j-th element from arr[]. For example, dp[1][1] = 3 because it means number
of subsequences of length 1 whose first element starts after 1-st element of arr[] which are {3}, {3}, {5}.
- Initialize first row of dp[][] to 1.
- Run two loops top to bottom and right to left inside of the former loop.
- If j > m – i that means there cannot be any such sequences due to lack of elements. So dp[i][j] = 0.
- Else, dp[i][j] = dp[i][j + 1] + arr[j] * dp[i – 1][j + 1] as the number will be number of already existing subsequences of length i plus the number of subsequences of length i – 1 multiplied by arr[j] due to repetition.

Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Returns number of subsequences
// of maximum length k and
// contains no repeated element
int countSubSeq(int a[], int n, int k)
{
    // Sort the array a[]
    sort(a, a + n);
    vector<int> arr;
 
    // Store the frequencies of all the
    // distinct element in the vector arr
    for (int i = 0; i < n;) {
        int count = 1, x = a[i];
        i++;
        while (i < n && a[i] == x) {
            count++;
            i++;
        }
        arr.push_back(count);
    }
 
    int m = arr.size();
    n = min(m, k);
 
    // count is the number
    // of such subsequences
    int count = 1;
 
    // Create a 2-d array dp[n+1][m+1] to
    // store the intermediate result
    int dp[n + 1][m + 1];
 
    // Initialize the first row to 1
    for (int i = 0; i <= m; i++)
        dp[0][i] = 1;
 
    // Update the dp[][] array based
    // on the recurrence relation
    for (int i = 1; i <= n; i++) {
        for (int j = m; j >= 0; j--) {
            if (j > m - i)
                dp[i][j] = 0;
            else {
                dp[i][j] = dp[i][j + 1]
                           + arr[j] * dp[i - 1][j + 1];
            }
        }
        count = count + dp[i][0];
    }
 
    // Return the number of subsequences
    return count;
}
 
// Driver code
int main()
{
    int a[] = { 2, 2, 3, 3, 5 };
    int n = sizeof(a) / sizeof(int);
    int k = 3;
 
    cout << countSubSeq(a, n, k);
 
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
 
class GFG 
{
 
// Returns number of subsequences
// of maximum length k and
// contains no repeated element
static int countSubSeq(int a[], int n, int k)
{
    // Sort the array a[]
    Arrays.sort(a);
    List<Integer> arr = new LinkedList<>();
 
    // Store the frequencies of all the
    // distinct element in the vector arr
    for (int i = 0; i < n;)
    {
        int count = 1, x = a[i];
        i++;
        while (i < n && a[i] == x) 
        {
            count++;
            i++;
        }
        arr.add(count);
    }
 
    int m = arr.size();
    n = Math.min(m, k);
 
    // count is the number
    // of such subsequences
    int count = 1;
 
    // Create a 2-d array dp[n+1][m+1] to
    // store the intermediate result
    int [][]dp = new int[n + 1][m + 1];
 
    // Initialize the first row to 1
    for (int i = 0; i <= m; i++)
        dp[0][i] = 1;
 
    // Update the dp[][] array based
    // on the recurrence relation
    for (int i = 1; i <= n; i++) 
    {
        for (int j = m; j >= 0; j--) 
        {
            if (j > m - i)
                dp[i][j] = 0;
            else
            {
                dp[i][j] = dp[i][j + 1] + 
                             arr.get(j) * 
                           dp[i - 1][j + 1];
            }
        }
        count = count + dp[i][0];
    }
 
    // Return the number of subsequences
    return count;
}
 
// Driver code
public static void main(String[] args) 
{
    int a[] = { 2, 2, 3, 3, 5 };
    int n = a.length;
    int k = 3;
 
    System.out.println(countSubSeq(a, n, k));
}
}
 
// This code is contributed by PrinciRaj1992

Python 3

# Python 3 implementation of the approach
 
# Returns number of subsequences
# of maximum length k and
# contains no repeated element
def countSubSeq(a, n, k):
     
    # Sort the array a[]
    a.sort(reverse = False)
    arr = []
 
    # Store the frequencies of all the
    # distinct element in the vector arr
    i = 0
    while(i < n):
        count = 1
        x = a[i]
        i += 1
        while (i < n and a[i] == x):
            count += 1
            i += 1
         
        arr.append(count)
 
    m = len(arr)
    n = min(m, k)
 
    # count is the number
    # of such subsequences
    count = 1
 
    # Create a 2-d array dp[n+1][m+1] to
    # store the intermediate result
    dp = [[0 for i in range(m + 1)] 
             for j in range(n + 1)]
 
    # Initialize the first row to 1
    for i in range(m + 1):
        dp[0][i] = 1
 
    # Update the dp[][] array based
    # on the recurrence relation
    for i in range(1, n + 1, 1):
        j = m
        while(j >= 0):
            if (j > m - i):
                dp[i][j] = 0
            else:
                dp[i][j] = dp[i][j + 1] + \
                  arr[j] * dp[i - 1][j + 1]
                 
            j -= 1
             
        count = count + dp[i][0]
 
    # Return the number of subsequences
    return count
 
# Driver code
if __name__ == '__main__':
    a = [2, 2, 3, 3, 5]
    n = len(a)
    k = 3
 
    print(countSubSeq(a, n, k))
 
# This code is contributed by Surendra_Gangwar

C#

// C# implementation of the approach
using System;
using System.Collections.Generic;
     
class GFG 
{
 
// Returns number of subsequences
// of maximum length k and
// contains no repeated element
static int countSubSeq(int []a, int n, int k)
{
    // Sort the array a[]
    Array.Sort(a);
    List<int> arr = new List<int>();
    int count, x;
     
    // Store the frequencies of all the
    // distinct element in the vector arr
    for (int i = 0; i < n;)
    {
        count = 1;
        x = a[i];
        i++;
        while (i < n && a[i] == x) 
        {
            count++;
            i++;
        }
        arr.Add(count);
    }
 
    int m = arr.Count;
    n = Math.Min(m, k);
 
    // count is the number
    // of such subsequences
    count = 1;
 
    // Create a 2-d array dp[n+1][m+1] to
    // store the intermediate result
    int [,]dp = new int[n + 1, m + 1];
 
    // Initialize the first row to 1
    for (int i = 0; i <= m; i++)
        dp[0, i] = 1;
 
    // Update the dp[][] array based
    // on the recurrence relation
    for (int i = 1; i <= n; i++) 
    {
        for (int j = m; j >= 0; j--) 
        {
            if (j > m - i)
                dp[i, j] = 0;
            else
            {
                dp[i, j] = dp[i, j + 1] + 
                                 arr[j] * 
                           dp[i - 1, j + 1];
            }
        }
        count = count + dp[i, 0];
    }
 
    // Return the number of subsequences
    return count;
}
 
// Driver code
public static void Main(String[] args) 
{
    int []a = { 2, 2, 3, 3, 5 };
    int n = a.Length;
    int k = 3;
 
    Console.WriteLine(countSubSeq(a, n, k));
}
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
 
// Javascript implementation of the approach
 
// Returns number of subsequences
// of maximum length k and
// contains no repeated element
function countSubSeq(a, n, k)
{
    // Sort the array a[]
    a.sort();
    var arr = [];
 
    // Store the frequencies of all the
    // distinct element in the vector arr
    for (var i = 0; i < n;) {
        var count = 1, x = a[i];
        i++;
        while (i < n && a[i] == x) {
            count++;
            i++;
        }
        arr.push(count);
    }
 
    var m = arr.length;
    n = Math.min(m, k);
 
    // count is the number
    // of such subsequences
    var count = 1;
 
    // Create a 2-d array dp[n+1][m+1] to
    // store the intermediate result
    var dp = Array.from(Array(n+1), ()=>Array(m+1));
 
    // Initialize the first row to 1
    for (var i = 0; i <= m; i++)
        dp[0][i] = 1;
 
    // Update the dp[][] array based
    // on the recurrence relation
    for (var i = 1; i <= n; i++) {
        for (var j = m; j >= 0; j--) {
            if (j > m - i)
                dp[i][j] = 0;
            else {
                dp[i][j] = dp[i][j + 1]
                           + arr[j] * dp[i - 1][j + 1];
            }
        }
        count = count + dp[i][0];
    }
 
    // Return the number of subsequences
    return count;
}
 
// Driver code
var a = [2, 2, 3, 3, 5];
var n = a.length;
var k = 3;
document.write( countSubSeq(a, n, k));
 
 
</script>

Output

Time Complexity: O(n*log(n)+n*m) where m is the size of the array and n=min(m,k).
Auxiliary Space: O(n*m)

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Returns number of subsequences
// of maximum length k and
// contains no repeated element
int countSubSeq(int a[], int n, int k)
{
    // Sort the array a[]
    sort(a, a + n);
    vector<int> arr;
 
    // Store the frequencies of all the
    // distinct element in the vector arr
    for (int i = 0; i < n;) {
        int count = 1, x = a[i];
        i++;
        while (i < n && a[i] == x) {
            count++;
            i++;
        }
        arr.push_back(count);
    }
 
    // count is the number
    // of such subsequences
    int m = arr.size();
    n = min(m, k);
 
    int count = 1;
 
    // to store computations of subproblems
    vector<int> dp(m + 1, 1);
 
    // iterate over subproblems to
    // get the current solution from previous computations
    for (int i = 1; i <= n; i++) {
 
        // vector to store current values
        vector<int> curr(m + 1, 0);
        for (int j = m; j >= 0; j--) {
            if (j > m - i)
                curr[j] = 0;
            else {
                curr[j] = curr[j + 1] + arr[j] * dp[j + 1];
            }
        }
 
        // assigning values ot iterate further
        dp = curr;
 
        // update count
        count = count + dp[0];
    }
 
    // return final answer
    return count;
}
 
// Driver code
int main()
{
    int a[] = { 2, 2, 3, 3, 5 };
    int n = sizeof(a) / sizeof(int);
    int k = 3;
 
    cout << countSubSeq(a, n, k);
 
    return 0;
}

Java

import java.util.Arrays;
 
public class GFG {
 
    public static int countSubSeq(int[] a, int k) {
        // Sort the input array in ascending order
        Arrays.sort(a);
        int[] arr = new int[a.length];
 
        // Count the occurrences of each unique element in the sorted array
        int m = 0;
        for (int i = 0; i < a.length; i++) {
            int count = 1;
            int x = a[i];
            while (i + 1 < a.length && a[i + 1] == x) {
                count++;
                i++;
            }
            arr[m++] = count;
        }
 
        m = Math.min(m, k);
        int count = 1;
        int[] dp = new int[m + 1];
        Arrays.fill(dp, 1);
 
        // Dynamic programming to calculate the count of subsequences
        for (int i = 1; i <= m; i++) {
            int[] curr = new int[m + 1];
            for (int j = m; j >= 0; j--) {
                if (j > m - i) {
                    curr[j] = 0;
                } else {
                    curr[j] = curr[j + 1] + arr[j] * dp[j + 1];
                }
            }
            dp = curr;
            count += dp[0];
        }
 
        return count;
    }
 
    public static void main(String[] args) {
        int[] a = {2, 2, 3, 3, 5};  // Given input array
        int k = 3;  // Maximum number of elements in a subsequence
 
        // Calculate and print the total count of subsequences with at most 'k' elements
        System.out.println(countSubSeq(a, k));
    }
}

Python

def countSubSeq(a, k):
    """
    Function to count the number of subsequences of an array 'a' with at most 'k' elements.
 
    Args:
        a (list): The input list containing the elements.
        k (int): The maximum number of elements in a subsequence.
 
    Returns:
        int: The total count of subsequences with at most 'k' elements.
    """
    a.sort()  # Sort the input list in ascending order
    arr = []
 
    # Count the occurrences of each unique element in the sorted array
    i = 0
    while i < len(a):
        count = 1
        x = a[i]
        i += 1
        while i < len(a) and a[i] == x:
            count += 1
            i += 1
        arr.append(count)
 
    m = len(arr)
    n = min(m, k)
 
    count = 1
    dp = [1] * (m + 1)
 
    # Dynamic programming to calculate the count of subsequences
    for i in range(1, n + 1):
        curr = [0] * (m + 1)
        for j in range(m, -1, -1):
            if j > m - i:
                curr[j] = 0
            else:
                curr[j] = curr[j + 1] + arr[j] * dp[j + 1]
        dp = curr
        count += dp[0]
 
    return count
 
 
# Driver code
a = [2, 2, 3, 3, 5]  # Given input list
k = 3  # Maximum number of elements in a subsequence
 
# Calculate and print the total count of subsequences with at most 'k' elements
print(countSubSeq(a, k))
#user_dtewbxkn77n

C#

using System;
 
public class GFG {
    public static int CountSubSeq(int[] a, int k)
    {
        // Sort the input array in ascending order
        Array.Sort(a);
        int[] arr = new int[a.Length];
 
        // Count the occurrences of each unique element in
        // the sorted array
        int m = 0;
        for (int i = 0; i < a.Length; i++) {
            int count = 1;
            int x = a[i];
            while (i + 1 < a.Length && a[i + 1] == x) {
                count++;
                i++;
            }
            arr[m++] = count;
        }
 
        m = Math.Min(m, k);
        int totalCount
            = 1; // Renamed 'count' to 'totalCount'
        int[] dp = new int[m + 1];
        Array.Fill(dp, 1);
 
        // Dynamic programming to calculate the count of
        // subsequences
        for (int i = 1; i <= m; i++) {
            int[] curr = new int[m + 1];
            for (int j = m; j >= 0; j--) {
                if (j > m - i) {
                    curr[j] = 0;
                }
                else {
                    curr[j]
                        = curr[j + 1] + arr[j] * dp[j + 1];
                }
            }
            dp = curr;
            totalCount
                += dp[0]; // Renamed 'count' to 'totalCount'
        }
 
        return totalCount; // Renamed 'count' to
                           // 'totalCount'
    }
 
    public static void Main()
    {
        int[] a = { 2, 2, 3, 3, 5 }; // Given input array
        int k = 3; // Maximum number of elements in a
                   // subsequence
 
        // Calculate and print the total count of
        // subsequences with at most 'k' elements
        Console.WriteLine(CountSubSeq(a, k));
    }
}

Javascript

// Returns number of subsequences
// of maximum length k and
// contains no repeated element
function countSubSeq(a, n, k) {
    // Sort the array a[]
    a.sort((x, y) => x - y);
    const arr = [];
 
    // Store the frequencies of all the
    // distinct element in the array arr
    let i = 0;
    while (i < n) {
        let count = 1;
        const x = a[i];
        i++;
        while (i < n && a[i] === x) {
            count++;
            i++;
        }
        arr.push(count);
    }
 
    // count is the number
    // of such subsequences
    let m = arr.length;
    n = Math.min(m, k);
 
    let count = 1;
 
    // to store computations of subproblems
    const dp = new Array(m + 1).fill(1);
 
    // iterate over subproblems to
    // get the current solution from previous computations
    for (let i = 1; i <= n; i++) {
 
        // array to store current values
        const curr = new Array(m + 1).fill(0);
        for (let j = m; j >= 0; j--) {
            if (j > m - i)
                curr[j] = 0;
            else {
                curr[j] = curr[j + 1] + arr[j] * dp[j + 1];
            }
        }
 
        // assigning values to iterate further
        dp.splice(0, dp.length, ...curr);
 
        // update count
        count = count + dp[0];
    }
 
    // return final answer
    return count;
}
 
// Driver code
const a = [2, 2, 3, 3, 5];
const n = a.length;
const k = 3;
 
console.log(countSubSeq(a, n, k));

Output

Time Complexity: O(n*log(n)+n*m) where m is the size of the array and n=min(m,k).
Auxiliary Space: O(m)

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