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Longest sub-sequence that satisfies the given conditions
  • Difficulty Level : Hard
  • Last Updated : 12 Jun, 2019

Given an array arr[] of N integers, the task is to find the longest sub-sequence in the given array such that for all pairs from the sub-sequence (arr[i], arr[j]) where i != j either arr[i] divides arr[j] or vice versa. If no such sub-sequence exists then print -1.

Examples:

Input: arr[] = {2, 4, 6, 1, 3, 11}
Output: 3
Longest valid sub-sequences are {1, 2, 6} and {1, 3, 6}.

Input: arr[] = {21, 22, 6, 4, 13, 7, 332}
Output: 2

Approach: This problem is a simple variation of the longest increasing sub-sequence problem. What changes is the base condition and the trick to reduce the number of computations by sorting the given array.



  • First sort the given array so that we only need to check values where arr[i] > arr[j] for i > j.
  • Then we move forward using two loops, outer loop runs from 1 to N and the inner loop runs from 0 to i.
  • Now in the inner loop we have to find the number of arr[j] where j is from 0 to i – 1 which divides the element arr[i].
  • And the recurrence relation will be dp[i] = max(dp[i], 1 + dp[j]).
  • We will update the max dp[i] value in a variable named res which will be the final answer.

Below is the implementation of the above approach:

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the length of the
// longest required sub-sequence
int find(int n, int a[])
{
    // Sort the array
    sort(a, a + n);
  
    // To store the resultant length
    int res = 1;
    int dp[n];
  
    // If array contains only one element
    // then it divides itself
    dp[0] = 1;
  
    for (int i = 1; i < n; i++) {
  
        // Every elemnet divides itself
        dp[i] = 1;
  
        // Count for all numbers which
        // are leser than a[i]
        for (int j = 0; j < i; j++) {
  
            if (a[i] % a[j] == 0) {
  
                // If a[i] % a[j] then update the maximum
                // subsequence length,
                // dp[i] = max(dp[i], 1 + dp[j])
                // where j is in the range [0, i-1]
                dp[i] = std::max(dp[i], 1 + dp[j]);
            }
        }
  
        res = std::max(res, dp[i]);
    }
  
    // If array contains only one element
    // then i = j which doesn't satisfy the condition
    return (res == 1) ? -1 : res;
}
  
// Driver code
int main()
{
    int a[] = { 2, 4, 6, 1, 3, 11 };
    int n = sizeof(a) / sizeof(int);
  
    cout << find(n, a);
  
    return 0;
}

Java




// Java implementation of the approach
import java.util.Arrays; 
import java.io.*;
  
class GFG 
{
      
// Function to return the length of the
// longest required sub-sequence
static int find(int n, int a[])
{
    // Sort the array
    Arrays.sort(a);
  
    // To store the resultant length
    int res = 1;
    int dp[] = new int[n];
  
    // If array contains only one element
    // then it divides itself
    dp[0] = 1;
  
    for (int i = 1; i < n; i++) 
    {
  
        // Every elemnet divides itself
        dp[i] = 1;
  
        // Count for all numbers which
        // are leser than a[i]
        for (int j = 0; j < i; j++) 
        {
  
            if (a[i] % a[j] == 0)
            {
  
                // If a[i] % a[j] then update the maximum
                // subsequence length,
                // dp[i] = Math.max(dp[i], 1 + dp[j])
                // where j is in the range [0, i-1]
                dp[i] = Math.max(dp[i], 1 + dp[j]);
            }
        }
  
        res = Math.max(res, dp[i]);
    }
  
    // If array contains only one element
    // then i = j which doesn't satisfy the condition
    return (res == 1) ? -1 : res;
}
  
// Driver code
public static void main (String[] args) 
{
    int a[] = { 2, 4, 6, 1, 3, 11 };
    int n = a.length;
  
    System.out.println (find(n, a));
}
}
  
// This code is contributed by jit_t

Python3




# Python3 implementation of the approach 
  
# Function to return the length of the 
# longest required sub-sequence 
def find(n, a) :
  
    # Sort the array 
    a.sort(); 
  
    # To store the resultant length 
    res = 1
    dp = [0]*n; 
  
    # If array contains only one element 
    # then it divides itself 
    dp[0] = 1
  
    for i in range(1, n) :
          
        # Every elemnet divides itself 
        dp[i] = 1
  
        # Count for all numbers which 
        # are leser than a[i] 
        for j in range(i) :
  
            if (a[i] % a[j] == 0) :
  
                # If a[i] % a[j] then update the maximum 
                # subsequence length, 
                # dp[i] = max(dp[i], 1 + dp[j]) 
                # where j is in the range [0, i-1] 
                dp[i] = max(dp[i], 1 + dp[j]); 
      
        res = max(res, dp[i]); 
  
    # If array contains only one element 
    # then i = j which doesn't satisfy the condition 
    if (res == 1): 
        return -1
    else :
        return res; 
  
  
# Driver code 
if __name__ == "__main__"
  
    a = [ 2, 4, 6, 1, 3, 11 ];
    n = len(a); 
  
    print(find(n, a)); 
      
# This code is contributed by AnkitRai01

C#




// C# implementation of the approach
using System;
  
class GFG 
{
      
// Function to return the length of the
// longest required sub-sequence
static int find(int n, int []a)
{
    // Sort the array
    Array.Sort(a);
  
    // To store the resultant length
    int res = 1;
    int []dp = new int[n];
  
    // If array contains only one element
    // then it divides itself
    dp[0] = 1;
  
    for (int i = 1; i < n; i++) 
    {
  
        // Every elemnet divides itself
        dp[i] = 1;
  
        // Count for all numbers which
        // are leser than a[i]
        for (int j = 0; j < i; j++) 
        {
  
            if (a[i] % a[j] == 0)
            {
  
                // If a[i] % a[j] then update the maximum
                // subsequence length,
                // dp[i] = Math.max(dp[i], 1 + dp[j])
                // where j is in the range [0, i-1]
                dp[i] = Math.Max(dp[i], 1 + dp[j]);
            }
        }
  
        res = Math.Max(res, dp[i]);
    }
  
    // If array contains only one element
    // then i = j which doesn't satisfy the condition
    return (res == 1) ? -1 : res;
}
  
// Driver code
public static void Main () 
{
    int []a = { 2, 4, 6, 1, 3, 11 };
    int n = a.Length;
  
    Console.WriteLine(find(n, a));
}
}
  
// This code is contributed by anuj_67..
Output:
3

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