Length of longest increasing index dividing subsequence

Given an array arr[] of size N, the task is to find the longest increasing sub-sequence such that index of any element is divisible by index of previous element (LIIDS). The following are the necessary conditions for the LIIDS:
If i, j are two indices in the given array. Then: 

• i < j
• j % i = 0
• arr[i] < arr[j]

Examples:

Input: arr[] = {1, 2, 3, 7, 9, 10} 
Output: 3 
Explanation:
The subsequence = {1, 2, 7} 
Indices of the elements of sub-sequence = {1, 2, 4} 
Indices condition: 
2 is divisible by 1 
4 is divisible by 2 
OR 
Another possible sub-sequence = {1, 3, 10} 
Indices of elements of sub-sequence = {1, 3, 6} 
Indices condition: 
3 is divisible by 1 
6 is divisible by 3
Input: arr[] = {7, 1, 3, 4, 6} 
Output: 2 
Explanation:
The sub-sequence = {1, 4} 
Indices of elements of sub-sequence = {2, 4} 
Indices condition: 
4 is divisible by 2 

Approach: The idea is to use the concept of Dynamic programming to solve this problem. 

• Create a dp[] array first and initialize the array with 1. This is because, the minimum length of the dividing subsequence is 1.
• Now, for every index ‘i’ in the array, increment the count of the values at the indices at all its multiples.
• Finally, when the above step is performed for all the values, the maximum value present in the array is the required answer.

Below is the implementation of the above approach:

C++

// C++ program to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element

#include <bits/stdc++.h>
using namespace std;

// Function to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element
int LIIDS(int arr[], int N)
{
    // Initialize the dp[] array with 1 as a
    // single element will be of 1 length
    int dp[N + 1];

    int ans = 0;
    for (int i = 1; i <= N; i++) {
        dp[i] = 1;
    }

    // Traverse the given array
    for (int i = 1; i <= N; i++) {

        // If the index is divisible by
        // the previous index
        for (int j = i + i; j <= N; j += i) {

            // if increasing
            // subsequence identified
            if (arr[j] > arr[i]) {
                dp[j] = max(dp[j], dp[i] + 1);
            }
        }

        // Longest length is stored
        ans = max(ans, dp[i]);
    }
    return ans;
}

// Driver code
int main()
{

    int arr[] = { 1, 2, 3, 7, 9, 10 };
    int N = sizeof(arr) / sizeof(arr[0]);

    cout << LIIDS(arr, N);
    return 0;
}

Java

// Java program to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element
import java.util.*;

class GFG{

// Function to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element
static int LIIDS(int arr[], int N)
{

    // Initialize the dp[] array with 1 as a
    // single element will be of 1 length
    int[] dp = new int[N + 1];
    int ans = 0;

    for(int i = 1; i <= N; i++)
    {
       dp[i] = 1;
    }

    // Traverse the given array
    for(int i = 1; i <= N; i++)
    {

       // If the index is divisible by
       // the previous index
       for(int j = i + i; j <= N; j += i)
       {

          // If increasing
          // subsequence identified
          if (j < arr.length && arr[j] > arr[i])
          {
              dp[j] = Math.max(dp[j], dp[i] + 1);
          }
       }

       // Longest length is stored
       ans = Math.max(ans, dp[i]);
    }
    return ans;
}

// Driver code
public static void main(String args[])
{
    int arr[] = { 1, 2, 3, 7, 9, 10 };
    int N = arr.length;

    System.out.println(LIIDS(arr, N));
}
}

// This code is contributed by AbhiThakur

Python3

# Python3 program to find the length of
# the longest increasing sub-sequence
# such that the index of the element is
# divisible by index of previous element

# Function to find the length of
# the longest increasing sub-sequence
# such that the index of the element is
# divisible by index of previous element
def LIIDS(arr, N):

    # Initialize the dp[] array with 1 as a
    # single element will be of 1 length
    dp = []
    for i in range(1, N + 1):
        dp.append(1)

    ans = 0

    # Traverse the given array
    for i in range(1, N + 1):

        # If the index is divisible by
        # the previous index
        j = i + i
        while j <= N:

            # If increasing
            # subsequence identified
            if j < N and i < N and arr[j] > arr[i]:
                dp[j] = max(dp[j], dp[i] + 1)

            j += i

        # Longest length is stored
        if i < N:
            ans = max(ans, dp[i])

    return ans

# Driver code
arr = [ 1, 2, 3, 7, 9, 10 ]

N = len(arr)

print(LIIDS(arr, N))

# This code is contributed by ishayadav181

C#

// C# program to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element
using System;

class GFG{

// Function to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element
static int LIIDS(int[] arr, int N)
{

    // Initialize the dp[] array with 1 as a
    // single element will be of 1 length
    int[] dp = new int[N + 1];
    int ans = 0;

    for(int i = 1; i <= N; i++)
    {
        dp[i] = 1;
    }

    // Traverse the given array
    for(int i = 1; i <= N; i++)
    {

        // If the index is divisible by
        // the previous index
        for(int j = i + i; j <= N; j += i)
        {

            // If increasing
            // subsequence identified
            if (j < arr.Length && arr[j] > arr[i])
            {
                dp[j] = Math.Max(dp[j], dp[i] + 1);
            }
        }

        // Longest length is stored
        ans = Math.Max(ans, dp[i]);
    }
    return ans;
}

// Driver code
public static void Main()
{
    int[] arr = new int[] { 1, 2, 3, 7, 9, 10 };
    int N = arr.Length;

    Console.WriteLine(LIIDS(arr, N));
}
}

// This code is contributed by sanjoy_62

Javascript

<script>

// Javascript program to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element

// Function to find the length of
// the longest increasing sub-sequence
// such that the index of the element is
// divisible by index of previous element
function LIIDS(arr, N)
{

    // Initialize the dp[] array with 1 as a
    // single element will be of 1 length
    let dp = [];
    let ans = 0;

    for(let i = 1; i <= N; i++)
    {
       dp[i] = 1;
    }

    // Traverse the given array
    for(let i = 1; i <= N; i++)
    {

       // If the index is divisible by
       // the previous index
       for(let j = i + i; j <= N; j += i)
       {

          // If increasing
          // subsequence identified
          if (j < arr.length && arr[j] > arr[i])
          {
              dp[j] = Math.max(dp[j], dp[i] + 1);
          }
       }

       // Longest length is stored
       ans = Math.max(ans, dp[i]);
    }
    return ans;
}

// Driver code

    let arr = [ 1, 2, 3, 7, 9, 10 ];
    let N = arr.length;

    document.write(LIIDS(arr, N));

// This code is contributed by susmitakundugoaldanga.
</script>

Output:

3

Time Complexity: O(N log(N)), where N is the length of the array.