Longest Increasing Subsequence

Preview

1. What is the Longest Increasing Subsequence (LIS) of a sequence of numbers?
2. How can the length of the LIS of a given sequence be determined?
3. What are some common algorithms used to solve for the LIS of a sequence of numbers?
4. How can dynamic programming be applied to solve for the LIS of a sequence of numbers efficiently?
5. What are some real-world applications of finding the LIS of a sequence of numbers?

Explain

The longest increasing subsequence (LIS) is a fundamental problem in computer science and mathematics that entails finding the longest subsequence in a given array of numbers that is strictly increasing.

For example, consider the array [3, 4, -1, 0, 6, 2, 3] The LIS in the given array of numbers is [3, 4, 6], where the subsequence is strictly increasing in terms of index values. There are other increasing subsequences in the array like [3, 4, 6] and [0, 2, 3], but the length of these subsequences is not the longest possible length of increasing subsequence.

The problem is solved using dynamic programming techniques that find the length of the LIS at each index of the array, which is based on the previously calculated LIS lengths. Finally, the subsequence is then constructed by backtracking through the array.

Keypoint

1. A subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.

2. An increasing subsequence is a subsequence in which the elements are in increasing order.

3. The longest increasing subsequence (LIS) problem is to find the longest increasing subsequence in a given sequence.

4. The LIS problem can be solved using dynamic programming.

5. The basic idea of the dynamic programming approach is to maintain an array that stores the length of the longest increasing subsequence ending at each element of the sequence.

6. Initially, the length of the LIS for each element in the sequence is set to 1.

7. For each element in the sequence, we determine the length of the LIS ending at that element by checking the lengths of the LIS ending at all of the previous elements that are smaller than the current element.

8. The length of the LIS for the current element is then set to one plus the maximum of these lengths.

9. The final answer is the maximum value in the array that stores the length of the LIS ending at each element in the sequence.

Review

1. What is the longest increasing subsequence of the sequence 2, 5, 3, 7, 11, 8, 10?
   Answer: The longest increasing subsequence is 2, 3, 7, 8, 10.

2. What is the time complexity of the longest increasing subsequence algorithm using dynamic programming?
   Answer: The time complexity is O(n^2), where n is the length of the sequence.

3. Can the longest increasing subsequence of a sequence consist of just one element?
   Answer: Yes, the longest increasing subsequence of a sequence can consist of just one element.

4. How many longest increasing subsequences can a sequence have?
   Answer: A sequence can have multiple longest increasing subsequences, but they all have the same length.

5. What is the minimum number of elements that need to be removed from a sequence to make it an increasing sequence?
   Answer: The minimum number of elements that need to be removed from a sequence to make it an increasing sequence is equal to the length of the sequence minus the length of the longest increasing subsequence.