Lecture: Largest Divisible Subset (DP44)

Introduction

• Topic: Problem-solving on "Largest Divisible Subset" using Dynamic Programming (DP).
• Pre-requisite: Understanding of earlier problems DP41 & DP42 related to "Longest Increasing Subsequence".
• Definition Clarification:
  • Subsequence: Elements in the same order as they appear in the array.
  • Subset: Elements not required to be in order; any elements can be picked.
  • Divisible Subset: Every pair of elements in the subset should be divisible by each other.

Problem Explanation

• Given an array of distinct numbers, find the largest subset where any two numbers divide each other.
• Order does not matter for subset.
• Example: Subset = {16, 8, 4}; all pairs (16, 8), (16, 4), (8, 4) are divisible.
• Largest subset example: {1, 16, 8, 4}; all pairs are divisible. Adding 7 would break divisibility.

Approach and Thought Process

• Convert problem to relate with "Longest Increasing Subsequence".
• Sort array to simplify divisibility checks.
• Use dynamic programming similar to LIS but with divisibility check.
• Recognize pattern: If array is sorted, if a number is divisible by the previous, it will be divisible by all earlier elements.

Dynamic Programming Solution

• Modify LIS approach:
  • After sorting the array, check if arr[i] % arr[j] == 0 instead of increasing condition.
  • Use a DP array to track the length of divisible subsets.
  • Use hash array to trace back the largest divisible subset.

Code Explanation

• Sort the array first.
• Implement DP similar to LIS, change condition to check divisibility.
• Time Complexity: O(n^2) for nested loops + O(n) for path tracing = O(n^2).
• Space Complexity: O(n).

Conclusion

• The problem is solved by leveraging the approach used in "Longest Increasing Subsequence".
• Sorting the array simplifies divisibility calculations.
• Key takeaway: Small modifications in logic can adapt a known algorithm to solve related problems.

Closing Remarks

• Importance of understanding core algorithms to solve variations of problems.
• Encourage subscribing and liking the video for more content.

Note: The above approach is efficient for arrays with a reasonable number of elements considering the O(n^2) complexity.