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Length of longest common prime subsequence from two given arrays

  • Difficulty Level : Hard
  • Last Updated : 11 Feb, 2021

Given two arrays arr1[] and arr2[] of length N and M respectively, the task is to find the length of the longest common prime subsequence that can be obtained from the two given arrays.

Examples: 

Input: arr1[] = {1, 2, 3, 4, 5, 6, 7, 8, 9}, arr2[] = {2, 5, 6, 3, 7, 9, 8} 
Output:
Explanation: 
The longest common prime subsequence present in both the arrays is {2, 3, 5, 7}.

Input: arr1[] = {1, 3, 5, 7, 9}, arr2[] = {2, 4, 6, 8, 10} 
Output:
Explanation: 
In the above arrays, the prime subsequence of arr1[] is {1, 3, 5, 7} and arr2[] is {2}. Therefore, there is no common prime numbers which are present in both the arrays. Hence, the result is 0.

Naive Approach: The simplest idea is to consider all subsequences of arr1[] and check if all numbers in this subsequence are prime and appear in arr2[]. Then find the longest length of these subsequences.



Time Complexity: O(M * 2N
Auxiliary Space: O(N)

Efficient Approach: The idea is to find all the prime numbers from both the arrays and then find the longest common prime subsequence from them using Dynamic Programming. Follow the steps below to solve the problem: 

  1. Find all the prime numbers between the minimum element of the array and maximum element of the array using Sieve Of Eratosthenes algorithm.
  2. Store the sequence of primes from the arrays arr1[] and arr2[].
  3. Find the LCS of the two sequences of primes.

Below is the implementation of the above approach:

C++




// CPP implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the LCS
int recursion(vector<int> arr1,
              vector<int> arr2, int i,
              int j, map<pair<int, int>,
              int> dp)
{
    if (i >= arr1.size() or j >= arr2.size())
        return 0;
    pair<int, int> key = { i, j };
    if (arr1[i] == arr2[j])
        return 1
               + recursion(arr1, arr2,
                            i + 1, j + 1,
                           dp);
    if (dp.find(key) != dp.end())
        return dp[key];
 
    else
        dp[key] = max(recursion(arr1, arr2,
                                i + 1, j, dp),
                      recursion(arr1, arr2, i,
                                j + 1, dp));
    return dp[key];
}
 
// Function to generate
// all the possible
// prime numbers
vector<int> primegenerator(int n)
{
    int cnt = 0;
    vector<int> primes(n + 1, true);
    int p = 2;
    while (p * p <= n)
    {
        for (int i = p * p; i <= n; i += p)
            primes[i] = false;
        p += 1;
    }
    return primes;
}
 
// Function which returns the
// length of longest common
// prime subsequence
int longestCommonSubseq(vector<int> arr1,
                        vector<int> arr2)
{
 
    // Minimum element of
    // both arrays
    int min1 = *min_element(arr1.begin(),
                            arr1.end());
    int min2 = *min_element(arr2.begin(),
                            arr2.end());
 
    // Maximum element of
    // both arrays
    int max1 = *max_element(arr1.begin(),
                            arr1.end());
    int max2 = *max_element(arr2.begin(),
                            arr2.end());
 
    // Generating all primes within
    // the max range of arr1
    vector<int> a = primegenerator(max1);
 
    // Generating all primes within
    // the max range of arr2
    vector<int> b = primegenerator(max2);
 
    vector<int> finala;
    vector<int> finalb;
 
    // Store precomputed values
    map<pair<int, int>, int> dp;
 
    // Store all primes in arr1[]
    for (int i = min1; i <= max1; i++)
    {
        if (find(arr1.begin(), arr1.end(), i)
            != arr1.end()
            and a[i] == true)
            finala.push_back(i);
    }
 
    // Store all primes of arr2[]
    for (int i = min2; i <= max2; i++)
    {
        if (find(arr2.begin(), arr2.end(), i)
            != arr2.end()
            and b[i] == true)
            finalb.push_back(i);
    }
 
    // Calculating the LCS
    return recursion(finala, finalb, 0, 0, dp);
}
 
// Driver Code
int main()
{
    vector<int> arr1 = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
    vector<int> arr2 = { 2, 5, 6, 3, 7, 9, 8 };
     
    // Function Call
    cout << longestCommonSubseq(arr1, arr2);
}

Java




// JAVA implementation of the above approach
import java.util.*;
import java.io.*;
import java.math.*;
public class GFG
{
 
  // Function to calculate the LCS
  static int recursion(ArrayList<Integer> arr1,
                       ArrayList<Integer> arr2, int i,
                       int j, Map<ArrayList<Integer>,
                       Integer> dp)
  {
    if (i >= arr1.size() || j >= arr2.size())
      return 0;
    ArrayList<Integer> key = new ArrayList<>();
    key.add(i);
    key.add(j);
    if (arr1.get(i) == arr2.get(j))
      return 1 + recursion(arr1, arr2,
                           i + 1, j + 1,
                           dp);
    if (dp.get(key) != dp.get(dp.size()-1))
      return dp.get(key);
 
    else
      dp.put(key,Math.max(recursion(arr1, arr2,
                                    i + 1, j, dp),
                          recursion(arr1, arr2, i,
                                    j + 1, dp)));
    return dp.get(key);
  }
 
  // Function to generate
  // all the possible
  // prime numbers
  static ArrayList<Boolean> primegenerator(int n)
  {
    int cnt = 0;
    ArrayList<Boolean> primes = new ArrayList<>();
    for(int i = 0; i < n + 1; i++)
      primes.add(true);
    int p = 2;
    while (p * p <= n)
    {
      for (int i = p * p; i <= n; i += p)
        primes.set(i,false);
      p += 1;
    }
    return primes;
  }
 
  // Function that returns the Minimum element of an ArrayList
  static int min_element(ArrayList<Integer> al)
  {
    int min = Integer.MAX_VALUE;
    for(int i = 0; i < al.size(); i++)
    {
      min=Math.min(min, al.get(i));
    }
    return min;
  }
 
  // Function that returns the Minimum element of an ArrayList
  static int max_element(ArrayList<Integer> al)
  {
    int max = Integer.MIN_VALUE;
    for(int i = 0; i < al.size(); i++)
    {
      max = Math.max(max, al.get(i));
    }
    return max;
  }
 
  // Function which returns the
  // length of longest common
  // prime subsequence
  static int longestCommonSubseq(ArrayList<Integer> arr1,
                                 ArrayList<Integer> arr2)
  {
 
    // Minimum element of
    // both arrays
    int min1 = min_element(arr1);
    int min2 = min_element(arr2);
 
    // Maximum element of
    // both arrays
    int max1 = max_element(arr1);
    int max2 = max_element(arr2);
 
    // Generating all primes within
    // the max range of arr1
    ArrayList<Boolean> a = primegenerator(max1);
 
    // Generating all primes within
    // the max range of arr2
    ArrayList<Boolean> b = primegenerator(max2);
    ArrayList<Integer> finala = new ArrayList<>();
    ArrayList<Integer> finalb = new ArrayList<>();
 
    // Store precomputed values
    Map<ArrayList<Integer>,Integer> dp =
      new HashMap <ArrayList<Integer>,Integer> ();
 
    // Store all primes in arr1[]
    for (int i = min1; i <= max1; i++)
    {
      if (arr1.contains(i)
          && a.get(i) == true)
        finala.add(i);
    }
 
    // Store all primes of arr2[]
    for (int i = min2; i <= max2; i++)
    {
      if (arr2.contains(i)
          && b.get(i) == true)
        finalb.add(i);
    }
 
    // Calculating the LCS
    return recursion(finala, finalb, 0, 0, dp);
  }
 
  // Driver Code
  public static void main(String args[])
  {
    int a1[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
    int a2[] = { 2, 5, 6, 3, 7, 9, 8 };
 
    // Converting into list
    ArrayList<Integer> arr1 = new ArrayList<Integer>();
    for(int i = 0; i < a1.length; i++)
      arr1.add(a1[i]);
 
    ArrayList<Integer> arr2 = new ArrayList<Integer>();
    for(int i = 0; i < a2.length; i++)
      arr2.add(a2[i]);
 
    // Function Call
    System.out.println(longestCommonSubseq(arr1, arr2));
  }
}
 
// This code is contributed by jyoti369

Python3




# Python implementation of the above approach
 
# Function to calculate the LCS
 
def recursion(arr1, arr2, i, j, dp):
    if i >= len(arr1) or j >= len(arr2):
        return 0
    key = (i, j)
    if arr1[i] == arr2[j]:
        return 1 + recursion(arr1, arr2,
                             i + 1, j + 1, dp)
    if key in dp:
        return dp[key]
    else:
        dp[key] = max(recursion(arr1, arr2,
                                i + 1, j, dp),
                      recursion(arr1, arr2,
                                i, j + 1, dp))
    return dp[key]
 
# Function to generate
# all the possible
# prime numbers
 
 
def primegenerator(n):
    cnt = 0
    primes = [True for _ in range(n + 1)]
    p = 2
    while p * p <= n:
        for i in range(p * p, n + 1, p):
            primes[i] = False
        p += 1
    return primes
 
# Function which returns the
# length of longest common
# prime subsequence
 
 
def longestCommonSubseq(arr1, arr2):
 
    # Minimum element of
    # both arrays
    min1 = min(arr1)
    min2 = min(arr2)
 
    # Maximum element of
    # both arrays
    max1 = max(arr1)
    max2 = max(arr2)
 
    # Generating all primes within
    # the max range of arr1
    a = primegenerator(max1)
 
    # Generating all primes within
    # the max range of arr2
    b = primegenerator(max2)
 
    finala = []
    finalb = []
 
    # Store precomputed values
    dp = dict()
 
    # Store all primes in arr1[]
    for i in range(min1, max1 + 1):
        if i in arr1 and a[i] == True:
            finala.append(i)
 
    # Store all primes of arr2[]
    for i in range(min2, max2 + 1):
        if i in arr2 and b[i] == True:
            finalb.append(i)
 
    # Calculating the LCS
    return recursion(finala, finalb,
                     0, 0, dp)
 
 
# Driver Code
arr1 = [1, 2, 3, 4, 5, 6, 7, 8, 9]
arr2 = [2, 5, 6, 3, 7, 9, 8]
 
# Function Call
print(longestCommonSubseq(arr1, arr2))
Output
4

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

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