How can I effectively implement the Longest Increasing Subsequence algorithm in Python using dynamic programming?
I recently switched to I've been working on this all day and I've been banging my head against this for hours... I'm trying to implement the Longest Increasing Subsequence (LIS) algorithm in Python, and I'm running into performance optimization with larger datasets. Currently, I'm using a dynamic programming approach, but it seems to be slow when the input size exceeds 1000 elements. Here's the code I've written: ```python def longest_increasing_subsequence(nums): if not nums: return 0 dp = [1] * len(nums) for i in range(1, len(nums)): for j in range(i): if nums[i] > nums[j]: dp[i] = max(dp[i], dp[j] + 1) return max(dp) ``` I tested it with a sample input of 1000 random integers and it took too long to execute. I'm specifically looking to understand if there's a more efficient way to implement this algorithm, ideally with a time complexity better than O(n^2). I've read about using binary search along with dynamic programming to optimize it, but I'm uncertain how to integrate that into my existing solution without overcomplicating it. Additionally, when I run my current implementation, I'm getting a maximum recursion depth exceeded behavior when I accidentally pass an empty list. I should probably handle that case better too, but I'm more concerned about the performance right now. Can someone guide me through the optimization process and also suggest improvements for handling edge cases? What am I doing wrong? I'd love to hear your thoughts on this. My development environment is Linux. Am I missing something obvious?