Introduction

The Longest Increasing Subsequence (LIS) is a very important problem in Data Structures and Algorithms (DSA) and is frequently asked in coding interviews. If you are enrolled in a DSA course in Jaipur, mastering LIS will help you understand sequence optimization problems and advanced dynamic programming concepts.

LIS focuses on finding a subsequence where elements are strictly increasing.

What is Longest Increasing Subsequence?

The Longest Increasing Subsequence is the longest subsequence of an array such that:

• Elements are in increasing order
• Order of elements is maintained
• Elements are not necessarily contiguous

Example:
Array: [10, 9, 2, 5, 3, 7, 101, 18]

LIS = [2, 3, 7, 101]
Length = 4

Key Characteristics

• Subsequence, not substring
• Increasing order required
• Relative order must be maintained

DP Approach (Classic Solution)

Idea

For each element, find LIS ending at that element.

Recurrence Relation

dp[i]=1+max⁡(dp[j]) where j<i and arr[j]<arr[i]

Time Complexity (DP Approach)

O(n^2)

Optimized Approach (Binary Search)

Use binary search to optimize LIS.

Idea

• Maintain a temporary array
• Replace elements using binary search

Time Complexity (Optimized)

O(n log⁡ n)

Steps for Optimized Approach

1. Create an empty list
2. Traverse array
3. Insert element in correct position using binary search
4. Replace if needed
5. Length of list = LIS

LIS vs LCS

• LIS works on one array
• LCS works on two strings
• LIS focuses on increasing order
• LCS focuses on matching sequences

Real-World Applications

• Stock market trend analysis
• Sequence prediction
• Data analysis
• Machine learning preprocessing

Common Interview Questions

• Find length of LIS
• Print LIS
• Count number of LIS
• Maximum sum increasing subsequence

Advantages

• Efficient with optimized approach
• Useful for sequence problems
• Builds strong DP foundation

Limitations

• Complex to understand initially
• Requires optimization for large inputs

Best Practices

• Start with O(n²) solution
• Then optimize using binary search
• Practice variations
• Understand sequence logic

Summary

• LIS finds longest increasing subsequence
• Uses dynamic programming
• Optimized solution runs in O(n log n)
• Important for coding interviews
• Builds advanced DP understanding

FAQs

Q1. What is LIS in DSA?
It is the longest increasing subsequence in an array.

Q2. What is the time complexity of LIS?
O(n²) for DP, O(n log n) for optimized approach.

Q3. Is LIS contiguous?
No, it is a subsequence, not a subarray.

Q4. Can LIS be optimized?
Yes, using binary search.

Q5. Is LIS important for interviews?
Yes, it is a frequently asked problem.

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