How to Find the Length of the Longest Increasing Subsequence

TL;DR

To find the length of the longest increasing subsequence, use a recursive approach with dynamic programming. Start by generating subsequences, checking their order; however, this brute force method is inefficient. By employing memoization to store computed values, you can optimize the solution and reduce the time complexity from exponential to quadratic.

Transcript

hey everyone welcome back to take you forward today we will be starting with another pattern of dp which is the lis pattern or the longest increasing subsequence pattern now we have done some problems on subsequence if you remember well enough the dp 13 to dp 22 was based on uh dp on subsequences but this is slightly different than that okay now lo... Read More

Key Insights

• 🛟 A subsequence can be obtained by removing elements from a given sequence while preserving their relative order.
• 🪘 The longest increasing subsequence requires the elements to be in strictly increasing order.
• ⌛ The brute force approach of generating all subsequences and checking for the increasing pattern is inefficient due to its exponential time complexity.
• ⌛ A recursive solution, although more efficient, still has exponential time complexity.
• 🏪 Memoization can be used to optimize the recursive solution by storing previously computed values.

Questions & Answers

Q: What is the definition of a subsequence?

A subsequence is a sequence that can be obtained by removing zero or more elements from a given sequence, while preserving their relative order.

Q: What is the objective of finding the longest increasing subsequence?

The objective is to determine the length of the longest subsequence from a given array where the elements are in increasing order.

Q: What is the difference between the longest increasing subsequence and the longest common subsequence?

The longest increasing subsequence requires the elements to be in strictly increasing order, while the longest common subsequence allows for equal elements and focuses on finding the maximum length common to two sequences.

Q: What is the time complexity of the brute force approach for finding the longest increasing subsequence?

The brute force approach has a time complexity of 2^n, as it involves generating all subsequences, which increases exponentially with the size of the input array.

Key Insights:

• A subsequence can be obtained by removing elements from a given sequence while preserving their relative order.
• The longest increasing subsequence requires the elements to be in strictly increasing order.
• The brute force approach of generating all subsequences and checking for the increasing pattern is inefficient due to its exponential time complexity.
• A recursive solution, although more efficient, still has exponential time complexity.
• Memoization can be used to optimize the recursive solution by storing previously computed values.
• The optimized solution using memoization has a time complexity of O(n^2) and space complexity of O(n).

Summary & Key Takeaways

• The video introduces the concept of subsequences and defines what qualifies as a subsequence.

• The objective is to find the length of the longest increasing subsequence from a given array.

• The video discusses the brute force approach of generating all subsequences and checking for the increasing pattern, but highlights its inefficiency.

• A recursive solution is presented, where the problem is broken down into subproblems, but also mentions the exponential time complexity of this approach.