# 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:3Explanation:

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 3Input:arr[] = {7, 1, 3, 4, 6}Output:2Explanation:

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.