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Count of subsequences of length atmost K containing distinct prime elements
  • Difficulty Level : Hard
  • Last Updated : 22 Oct, 2020

Given an array arr of length N and an integer K, the task is to count the number of possible subsequences of length at most K which contains distinct prime elements from the array. 
Examples: 

Input: arr[] = {1, 2, 2, 3, 3, 4, 5}, N = 7, K = 3 
Output: 18 
Explanation: {}, {2}, {2}, {3}, {3}, {5}, {2, 3}, {2, 3}, {2, 3}, {2, 3}, {2, 5}, {2, 5}, {3, 5}, {3, 5}, {2, 3, 5}, {2, 3, 5}, {2, 3, 5} and {2, 3, 5} are the subsequences.
Input arr[] = {2, 4, 6, 7, 3, 9, 11, 5}, N = 8, K = 3 
Output: 26 

Approach: 
Using Sieve of Eratosthenes, precompute and store all prime numbers. Calculate and store the frequency of every prime number in the given array. Using a dynamic programming approach, calculate the number of subsequences of length 2 to K. Keep updating ans by adding the possible distinct combinations of length 2 to K for each dp[i]. Once calculated, add the frequency of all the primes + 1 as the subsequences of length 1 and 0. The final value of ans gives the desired result.
Below is the implementation of the above approach:
 

C++

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// C++ Program to find the
// count of distinct prime
// subsequences at most of
// of length K from a given array
 
#include <bits/stdc++.h>
using namespace std;
 
bool prime[100001];
 
void SieveOfEratosthenes()
{
    // Initialize all indices as true
    memset(prime, true, sizeof(prime));
 
    prime[0] = prime[1] = false;
 
    // A value in prime[i] will finally be
    // false if i is not a prime, else true
    for (int p = 2; p * p <= 100000; p++) {
 
        // If prime[p] is true,
        // then it is a prime
        if (prime[p] == true) {
 
            // Update all multiples of p
            // as false, i.e. non-prime
            for (int i = p * p;
                 i <= 100000;
                 i += p)
 
                prime[i] = false;
        }
    }
}
 
// Returns number of subsequences
// of maximum length k and
// contains distinct primes
int distinctPrimeSubSeq(
    int a[],
    int n, int k)
{
    SieveOfEratosthenes();
 
    // Store the primes in
    // the given array
    vector<int> primes;
 
    for (int i = 0; i < n; i++) {
        if (prime[a[i]])
            primes.push_back(a[i]);
    }
 
    int l = primes.size();
    // Sort the primes
    sort(primes.begin(),
         primes.end());
 
    // Store the frequencies
    // of all the
    // distinct primes
    vector<int> b;
    vector<int> dp;
    int sum = 0;
 
    for (int i = 0; i < l;) {
        int count = 1, x = a[i];
        i++;
        while (i < l && a[i] == x) {
            count++;
            i++;
        }
 
        // Store the frequency
        // of primes
        b.push_back(count);
        dp.push_back(count);
 
        // Store the sum of all
        // frequencies
        sum += count;
    }
 
    // Store the length of
    // subsequence at every
    // instant
    int of_length = 2;
    int len = dp.size();
    int ans = 0;
 
    while (of_length <= k) {
 
        // Store the frequency
        int freq = 0;
 
        // Store the previous
        // count of updated DP
        int prev = 0;
 
        for (int i = 0; i < (len - 1); i++) {
            freq += dp[i];
 
            int j = sum - freq;
 
            // Calculate total subsequences
            // of current of_length
            int subseq = b[i] * j;
 
            // Add the number of
            // subsequences to the answer
            ans += subseq;
 
            // Update the value in dp[i]
            dp[i] = subseq;
 
            // Store the updated dp[i]
            prev += dp[i];
        }
 
        len--;
        sum = prev;
        of_length++;
    }
 
    ans += (l + 1);
 
    return ans;
}
 
// Driver Code
int main()
{
    int a[] = { 1, 2, 2, 3, 3, 4, 5 };
    int n = sizeof(a) / sizeof(int);
    int k = 3;
 
    cout << distinctPrimeSubSeq(a, n, k);
 
    return 0;
}

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Java

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// Java Program to find the
// count of distinct prime
// subsequences at most of
// of length K from a given array
import java.util.*;
class GFG{
 
static boolean []prime =
       new boolean[100001];
 
static void SieveOfEratosthenes()
{
  // Initialize all indices as true
  for (int i = 0; i < prime.length; i++)
    prime[i] = true;
 
  prime[0] = prime[1] = false;
 
  // A value in prime[i] will finally
  // be false if i is not a prime,
  // else true
  for (int p = 2; p * p < 100000; p++)
  {
    // If prime[p] is true,
    // then it is a prime
    if (prime[p] == true)
    {
      // Update all multiples of p
      // as false, i.e. non-prime
      for (int i = p * p;
               i <= 100000; i += p)
        prime[i] = false;
    }
  }
}
 
// Returns number of subsequences
// of maximum length k and
// contains distinct primes
static int distinctPrimeSubSeq(int a[],
                               int n,
                               int k)
{
  SieveOfEratosthenes();
 
  // Store the primes in
  // the given array
  Vector<Integer> primes =
         new Vector<>();
 
  for (int i = 0; i < n; i++)
  {
    if (prime[a[i]])
      primes.add(a[i]);
  }
 
  int l = primes.size();
   
  // Sort the primes
  Collections.sort(primes);
 
  // Store the frequencies
  // of all the
  // distinct primes
  Vector<Integer> b =
         new Vector<>();
  Vector<Integer> dp =
         new Vector<>();
  int sum = 0;
 
  for (int i = 0; i < l;)
  {
    int count = 1, x = a[i];
    i++;
    while (i < l && a[i] == x)
    {
      count++;
      i++;
    }
 
    // Store the frequency
    // of primes
    b.add(count);
    dp.add(count);
 
    // Store the sum of all
    // frequencies
    sum += count;
  }
 
  // Store the length of
  // subsequence at every
  // instant
  int of_length = 2;
  int len = dp.size();
  int ans = 0;
 
  while (of_length < k)
  {
    // Store the frequency
    int freq = 0;
 
    // Store the previous
    // count of updated DP
    int prev = 0;
 
    for (int i = 0;
             i < (len - 1); i++)
    {
      freq += dp.elementAt(i);
 
      int j = sum - freq;
 
      // Calculate total subsequences
      // of current of_length
      int subseq = b.elementAt(i) * j;
 
      // Add the number of
      // subsequences to the answer
      ans += subseq;
 
      // Update the value in dp[i]
      dp.add(i, subseq);
 
      // Store the updated dp[i]
      prev += dp.elementAt(i);
    }
 
    len--;
    sum = prev;
    of_length++;
  }
  ans += (l + 3);
  return ans;
}
 
// Driver Code
public static void main(String[] args)
{
  int a[] = {1, 2, 2, 3, 3, 4, 5};
  int n = a.length;
  int k = 3;
  System.out.print(distinctPrimeSubSeq(a, n, k));
}
}
 
// This code is contributed by Princi Singh

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Python3

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# Python3 Program to find the
# count of distinct prime
# subsequences at most of
# of length K from a given array
 
prime = [True] * 1000000
 
def SieveOfEratosthenes():
 
    global prime
 
    prime[0] = prime[1] = False
 
    # A value in prime[i] will
    # finally be false if i is
    # not a prime, else true
    p = 2
     
    while p * p <= 100000:
 
        # If prime[p] is true,
        # then it is a prime
        if (prime[p] == True):
 
            # Update all multiples of p
            # as false, i.e. non-prime
            for i in range(p * p,
                           100001, p):
                prime[i] = False
                 
        p += 1
 
# Returns number of subsequences
# of maximum length k and
# contains distinct primes
def distinctPrimeSubSeq(a,
                        n, k):
 
    SieveOfEratosthenes()
 
    # Store the primes in
    # the given array
    primes = []
 
    for i in range(n):
        if (prime[a[i]]):
            primes.append(a[i])
 
    l = len(primes)
     
    # Sort the primes
    primes.sort()
 
    # Store the frequencies
    # of all the
    # distinct primes
    b = []
    dp = []
    sum = 0
 
    i = 0
    while i < l:
        count = 1
        x = a[i]
        i += 1
        while (i < l and
               a[i] == x):
            count += 1
            i += 1
 
        # Store the frequency
        # of primes
        b.append(count)
        dp.append(count)
 
        # Store the sum of all
        # frequencies
        sum += count
 
    # Store the length of
    # subsequence at every
    # instant
    of_length = 2
    leng = len(dp)
    ans = 0
 
    while (of_length <= k):
 
        # Store the frequency
        freq = 0
 
        # Store the previous
        # count of updated DP
        prev = 0
 
        for i in range(leng - 1):
            freq += dp[i]
 
            j = sum - freq
 
            # Calculate total subsequences
            # of current of_length
            subseq = b[i] * j
 
            # Add the number of
            # subsequences to the answer
            ans += subseq
 
            # Update the value in dp[i]
            dp[i] = subseq
 
            # Store the updated dp[i]
            prev += dp[i]
 
        leng -= 1
        sum = prev
        of_length += 1
 
    ans += (l + 1)
    return ans
 
# Driver Code
if __name__ == "__main__":
 
    a = [1, 2, 2,
         3, 3, 4, 5]
    n = len(a)
    k = 3
    print(distinctPrimeSubSeq(a, n, k))
 
# This code is contributed by Chitranayal

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C#

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// C# Program to find the
// count of distinct prime
// subsequences at most of
// of length K from a given array
using System;
using System.Collections.Generic;
class GFG{
 
static bool []prime =
       new bool[100001];
 
static void SieveOfEratosthenes()
{
  // Initialize all indices as true
  for (int i = 0; i < prime.Length; i++)
    prime[i] = true;
 
  prime[0] = prime[1] = false;
 
  // A value in prime[i] will finally
  // be false if i is not a prime,
  // else true
  for (int p = 2; p * p < 100000; p++)
  {
    // If prime[p] is true,
    // then it is a prime
    if (prime[p] == true)
    {
      // Update all multiples of p
      // as false, i.e. non-prime
      for (int i = p * p;
               i <= 100000; i += p)
        prime[i] = false;
    }
  }
}
 
// Returns number of subsequences
// of maximum length k and
// contains distinct primes
static int distinctPrimeSubSeq(int []a,
                               int n,
                               int k)
{
  SieveOfEratosthenes();
 
  // Store the primes in
  // the given array
  List<int> primes = new List<int>();
 
  for (int i = 0; i < n; i++)
  {
    if (prime[a[i]])
      primes.Add(a[i]);
  }
 
  int l = primes.Count;
   
  // Sort the primes
  primes.Sort();
 
  // Store the frequencies
  // of all the
  // distinct primes
  List<int> b = new List<int>();
  List<int> dp = new List<int>();
  int sum = 0;
 
  for (int i = 0; i < l;)
  {
    int count = 1, x = a[i];
    i++;
     
    while (i < l && a[i] == x)
    {
      count++;
      i++;
    }
 
    // Store the frequency
    // of primes
    b.Add(count);
    dp.Add(count);
 
    // Store the sum of all
    // frequencies
    sum += count;
  }
 
  // Store the length of
  // subsequence at every
  // instant
  int of_length = 2;
  int len = dp.Count;
  int ans = 0;
 
  while (of_length <= k)
  {
    // Store the frequency
    int freq = 0;
 
    // Store the previous
    // count of updated DP
    int prev = 0;
 
    for (int i = 0;
             i < (len ); i++)
    {
      freq += dp[i];
      int j = sum - freq;
 
      // Calculate total subsequences
      // of current of_length
      int subseq = b[i] * j;
 
      // Add the number of
      // subsequences to the answer
      ans += subseq;
 
      // Update the value in dp[i]
      dp[i] = subseq;
 
      // Store the updated dp[i]
      prev += dp[i];
    }
 
    len--;
    sum = prev;
    of_length++;
  }
   
  ans += (l + 1);
  return ans;
}
 
// Driver Code
public static void Main(String[] args)
{
  int []a = {1, 2, 2, 3, 3, 4, 5};
  int n = a.Length;
  int k = 3;
  Console.Write(distinctPrimeSubSeq(a, n, k));
}
}
 
// This code is contributed by Princi Singh

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Output: 

18







 

Time Complexity: O(K*(Distinct number of prime))

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