How would you find the longest increasing subsequence in an array?

Explanation:

To find the longest increasing subsequence (LIS) in an array, you are looking to identify the longest sequence of elements that are in strictly increasing order. This is a classic dynamic programming problem. The idea is to maintain, for each element in the array, the length of the longest increasing subsequence that ends with that element, and then build up to the solution for the entire array.

A common approach to solving this problem involves using dynamic programming with a time complexity of (O(n^2)), where (n) is the number of elements in the array. There is also a more optimized solution using a combination of dynamic programming and binary search with a time complexity of (O(n \log n)).

Key Talking Points:

• The problem is about finding the longest subsequence, not necessarily contiguous, that is strictly increasing.
• Dynamic programming can solve this in (O(n^2)) time.
• An optimized approach using dynamic programming and binary search can reduce the time complexity to (O(n \log n)).

NOTES:

Reference Table:

Pseudocode:

Here's a brief pseudocode for the (O(n \log n)) solution using dynamic programming and binary search:

def length_of_LIS(nums):
    if not nums:
        return 0

    lis = []
    for num in nums:
        pos = binary_search(lis, num)
        if pos == len(lis):
            lis.append(num)
        else:
            lis[pos] = num
    return len(lis)

def binary_search(lis, target):
    low, high = 0, len(lis)
    while low < high:
        mid = (low + high) // 2
        if lis[mid] < target:
            low = mid + 1
        else:
            high = mid
    return low

Follow-Up Questions and Answers:

1. Can you modify the algorithm to return the actual longest increasing subsequence, not just its length?

   • Answer: Yes, you can store additional information about the predecessors of each element and reconstruct the sequence at the end.

2. How would you handle finding the longest decreasing subsequence?

   • Answer: You could apply a similar approach by reversing the order condition to find the longest decreasing subsequence.

3. What if the input array is linked list or streamed data?

   • Answer: For a linked list, you would still follow a similar approach but would need to take care of list traversal instead of index-based access. For streamed data, you might need to maintain a dynamic list like the (O(n \log n)) approach, but you'll face challenges if the stream doesn't allow for multiple passes over the data.