The longest increasing subsequence (LIS) problem involves finding the longest subsequence within a sequence of numbers such that the subsequence is strictly increasing. One common approach to solve this problem is using dynamic programming (DP), which has a time complexity of O(n^2). In TypeScript, you can implement the LIS solution by maintaining an array `dp`, where `dp[i]` represents the length of the longest increasing subsequence ending at index `i`. As you iterate through the array, you update the `dp` array by checking for any previous elements that are smaller than the current one, and if found, you update the subsequence length. A more optimized solution uses binary search to maintain a running list of the longest subsequences, which reduces the time complexity to O(n log n). Understanding how to implement the LIS in TypeScript is useful in various real-world applications such as version control, ranking algorithms, and genetic sequence analysis. By mastering LIS, developers can apply these techniques in optimization problems where sequences and order play a critical role.