Longest subsequence such that every element in the subsequence is formed by multiplying previous element with a prime

Given a sorted array of N integers. The task is to find the longest subsequence such that every element in the subsequence is reachable by multiplying any prime number to the previous element in the subsequence.

Note: A[i] <= 10⁵

Examples:

Input: a[] = {3, 5, 6, 12, 15, 36}
Output 4
The longest subsequence is {3, 6, 12, 36}
6 = 3*2
12 = 6*2
36 = 12*3

2 and 3 are primes

Input: a[] = {1, 2, 5, 6, 12, 35, 60, 385}
Output: 5

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The problem can be solved using pre-storing primes till the largest number in the array and using basic Dynamic Programming. The following steps can be followed to solve the above problem:

Initially store all the primes in any of the Data-structure.
Hash the index of the numbers in a hash-map.
Create a dp[] of size N, and initialize it with 1 at every place, as the longest subsequence possible is 1 only. dp[i] represents the length of the longest subsequence that can be formed with a[i] as the starting element.
Iterate from the n-2, and for every number multiply it with all the primes till it exceeds a[n-1] and perform the below operations.
Multiply the number a[i] with prime to get x. If x exists in the hash-map then the recurrence will be dp[i] = max(dp[i], 1 + dp[hash[x]]).
At the end, iterate in dp[] array and find the maximum value which will be our answer.

Below is the implementation of the above approach:

C++

// C++ program to implement the 
// above approach 
#include <bits/stdc++.h> 
using namespace std; 
  
// Fucntion to pre-store primes 
void SieveOfEratosthenes(int MAX, vector<int>& primes) 
{ 
    bool prime[MAX + 1]; 
    memset(prime, true, sizeof(prime)); 
  
    // Sieve method to check if prime or not 
    for (long long p = 2; p * p <= MAX; p++) { 
        if (prime[p] == true) { 
            // Multiples 
            for (long long i = p * p; i <= MAX; i += p) 
                prime[i] = false; 
        } 
    } 
  
    // Pre-store all the primes 
    for (long long i = 2; i <= MAX; i++) { 
        if (prime[i]) 
            primes.push_back(i); 
    } 
} 
  
// Function to find the longest subsequence 
int findLongest(int A[], int n) 
{ 
    // Hash map 
    unordered_map<int, int> mpp; 
    vector<int> primes; 
  
    // Call the function to pre-store the primes 
    SieveOfEratosthenes(A[n - 1], primes); 
  
    int dp[n]; 
    memset(dp, 0, sizeof dp); 
  
    // Initialize last element with 1 
    // as that is the longest possible 
    dp[n - 1] = 1; 
    mpp[A[n - 1]] = n - 1; 
  
    // Iterate from the back and find the longest 
    for (int i = n - 2; i >= 0; i--) { 
  
        // Get the number 
        int num = A[i]; 
  
        // Initialize dp[i] as 1 
        // as the element will only me in 
        // the subsequence . 
        dp[i] = 1; 
        int maxi = 0; 
  
        // Iterate in all the primes and 
        // multiply to get the next element 
        for (auto it : primes) { 
  
            // Next element if multiplied with it 
            int xx = num * it; 
  
            // If exceeds the last element 
            // then break 
            if (xx > A[n - 1]) 
                break; 
  
            // If the number is there in the array 
            else if (mpp[xx] != 0) { 
                // Get the maximum most element 
                dp[i] = max(dp[i], 1 + dp[mpp[xx]]); 
            } 
        } 
        // Hash the element 
        mpp[A[i]] = i; 
    } 
    int ans = 1; 
  
    // Find the longest 
    for (int i = 0; i < n; i++) { 
        ans = max(ans, dp[i]); 
    } 
  
    return ans; 
} 
// Driver Code 
int main() 
{ 
    int a[] = { 1, 2, 5, 6, 12, 35, 60, 385 }; 
    int n = sizeof(a) / sizeof(a[0]); 
    cout << findLongest(a, n); 
} 

Java

// Java program to implement the  
// above approach 
import java.util.HashMap; 
import java.util.Vector; 
  
class GFG  
{ 
  
    // Fucntion to pre-store primes 
    public static void SieveOfEratosthenes(int MAX,  
                            Vector<Integer> primes)  
    { 
        boolean[] prime = new boolean[MAX + 1]; 
        for (int i = 0; i < MAX + 1; i++) 
            prime[i] = true; 
  
        // Sieve method to check if prime or not 
        for (int p = 2; p * p <= MAX; p++)  
        { 
            if (prime[p] == true) 
            { 
                // Multiples 
                for (int i = p * p; i <= MAX; i += p) 
                    prime[i] = false; 
            } 
        } 
  
        // Pre-store all the primes 
        for (int i = 2; i <= MAX; i++) 
        { 
            if (prime[i]) 
                primes.add(i); 
        } 
    } 
  
    // Function to find the intest subsequence 
    public static int findLongest(int[] A, int n) 
    { 
  
        // Hash map 
        HashMap<Integer, Integer> mpp = new HashMap<>(); 
        Vector<Integer> primes = new Vector<>(); 
  
        // Call the function to pre-store the primes 
        SieveOfEratosthenes(A[n - 1], primes); 
  
        int[] dp = new int[n]; 
  
        // Initialize last element with 1 
        // as that is the intest possible 
        dp[n - 1] = 1; 
        mpp.put(A[n - 1], n - 1); 
  
        // Iterate from the back and find the intest 
        for (int i = n - 2; i >= 0; i--) 
        { 
  
            // Get the number 
            int num = A[i]; 
  
            // Initialize dp[i] as 1 
            // as the element will only me in 
            // the subsequence . 
            dp[i] = 1; 
            int maxi = 0; 
  
            // Iterate in all the primes and 
            // multiply to get the next element 
            for (int it : primes)  
            { 
  
                // Next element if multiplied with it 
                int xx = num * it; 
  
                // If exceeds the last element 
                // then break 
                if (xx > A[n - 1]) 
                    break; 
  
                // If the number is there in the array 
                else if (mpp.get(xx) != null && mpp.get(xx) != 0)  
                { 
  
                        // Get the maximum most element 
                        dp[i] = Math.max(dp[i], 1 + dp[mpp.get(xx)]); 
                } 
            } 
  
            // Hash the element 
            mpp.put(A[i], i); 
        } 
        int ans = 1; 
  
        // Find the intest 
        for (int i = 0; i < n; i++) 
            ans = Math.max(ans, dp[i]); 
  
        return ans; 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        int[] a = { 1, 2, 5, 6, 12, 35, 60, 385 }; 
        int n = a.length; 
        System.out.println(findLongest(a, n)); 
  
    } 
} 
  
// This code is contributed by 
// sanjeev2552 

Python3

# Python3 program to implement the  
# above approach  
  
from math import sqrt 
  
# Fucntion to pre-store primes  
def SieveOfEratosthenes(MAX, primes) :  
  
    prime = [True]*(MAX + 1);  
  
    # Sieve method to check if prime or not  
    for p in range(2,int(sqrt(MAX)) + 1) :  
        if (prime[p] == True) : 
            # Multiples  
            for i in range(p**2, MAX + 1, p) :  
                prime[i] = False;  
  
    # Pre-store all the primes  
    for i in range(2, MAX + 1) : 
        if (prime[i]) : 
            primes.append(i);  
  
# Function to find the longest subsequence  
def findLongest(A, n) :  
  
    # Hash map  
    mpp = {}; 
    primes = []; 
      
    # Call the function to pre-store the primes 
    SieveOfEratosthenes(A[n - 1], primes); 
      
    dp = [0] * n ; 
      
    # Initialize last element with 1 
    # as that is the longest possible 
    dp[n - 1] = 1; 
    mpp[A[n - 1]] = n - 1; 
      
    # Iterate from the back and find the longest 
    for i in range(n-2,-1,-1) : 
          
        # Get the number 
        num = A[i]; 
          
        # Initialize dp[i] as 1 
        # as the element will only me in 
        # the subsequence  
        dp[i] = 1; 
        maxi = 0; 
          
        # Iterate in all the primes and 
        # multiply to get the next element 
        for it in primes : 
              
            # Next element if multiplied with it 
            xx = num * it; 
              
            # If exceeds the last element 
            # then break 
            if (xx > A[n - 1]) : 
                break; 
                  
            # If the number is there in the array 
            elif xx in mpp : 
                # Get the maximum most element 
                dp[i] = max(dp[i], 1 + dp[mpp[xx]]); 
                  
        # Hash the element 
        mpp[A[i]] = i; 
          
    ans = 1; 
          
    # Find the longest 
    for i in range(n) : 
        ans = max(ans, dp[i]); 
          
    return ans;  
  
# Driver Code  
if __name__ == "__main__" :  
  
    a = [ 1, 2, 5, 6, 12, 35, 60, 385 ];  
    n = len(a);  
      
    print(findLongest(a, n));  
  
# This code is contributed by AnkitRai01 

C#

// C# program to implement the  
// above approach 
using System; 
using System.Collections.Generic;  
  
class GFG  
{ 
  
    // Fucntion to pre-store primes 
    public static void SieveOfEratosthenes(int MAX,  
                                      List<int> primes)  
    { 
        Boolean[] prime = new Boolean[MAX + 1]; 
        for (int i = 0; i < MAX + 1; i++) 
            prime[i] = true; 
  
        // Sieve method to check if prime or not 
        for (int p = 2; p * p <= MAX; p++)  
        { 
            if (prime[p] == true) 
            { 
                // Multiples 
                for (int i = p * p; i <= MAX; i += p) 
                    prime[i] = false; 
            } 
        } 
  
        // Pre-store all the primes 
        for (int i = 2; i <= MAX; i++) 
        { 
            if (prime[i]) 
                primes.Add(i); 
        } 
    } 
  
    // Function to find the intest subsequence 
    public static int findLongest(int[] A, int n) 
    { 
  
        // Hash map 
        Dictionary<int, int> mpp = new Dictionary<int, int>(); 
        List<int> primes = new List<int>(); 
  
        // Call the function to pre-store the primes 
        SieveOfEratosthenes(A[n - 1], primes); 
  
        int[] dp = new int[n]; 
  
        // Initialize last element with 1 
        // as that is the intest possible 
        dp[n - 1] = 1; 
        mpp.Add(A[n - 1], n - 1); 
  
        // Iterate from the back and find the intest 
        for (int i = n - 2; i >= 0; i--) 
        { 
  
            // Get the number 
            int num = A[i]; 
  
            // Initialize dp[i] as 1 
            // as the element will only me in 
            // the subsequence . 
            dp[i] = 1; 
  
            // Iterate in all the primes and 
            // multiply to get the next element 
            foreach (int it in primes)  
            { 
  
                // Next element if multiplied with it 
                int xx = num * it; 
  
                // If exceeds the last element 
                // then break 
                if (xx > A[n - 1]) 
                    break; 
  
                // If the number is there in the array 
                else if (mpp.ContainsKey(xx) && mpp[xx] != 0)  
                { 
  
                    // Get the maximum most element 
                    dp[i] = Math.Max(dp[i], 1 + dp[mpp[xx]]); 
                } 
            } 
  
            // Hash the element 
            if(mpp.ContainsKey(A[i])) 
                mpp[A[i]] = i; 
            else
                mpp.Add(A[i], i); 
        } 
        int ans = 1; 
  
        // Find the intest 
        for (int i = 0; i < n; i++) 
            ans = Math.Max(ans, dp[i]); 
  
        return ans; 
    } 
  
    // Driver code 
    public static void Main(String[] args) 
    { 
        int[] a = { 1, 2, 5, 6, 12, 35, 60, 385 }; 
        int n = a.Length; 
        Console.WriteLine(findLongest(a, n)); 
    } 
} 
  
// This code is contributed by Rajput-Ji 

Output:

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

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

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