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Asked: 2024-09-25T15:46:17+05:30

Finding the Longest Alternating Subsequence: A Challenge of Sign and Order

I’ve been diving into some sequence problems lately, and I stumbled upon this fascinating challenge that I can’t quite wrap my head around. I thought it would be fun to share and see how everyone else tackles it!

The task revolves around finding the longest alternating subsequence in a given sequence of integers. The twist? The integers can be positive, negative, or zero. For those unfamiliar, an alternating subsequence means that the elements consistently switch from one sign to another. So, you’re looking for a sequence where each number is either less than or greater than the one preceding it—that is, they can’t all be increasing or decreasing continuously.

Here’s the kicker: I’ve seen some cases where the average sequences seem pretty straightforward, but then there are those tricky arrangements with lots of back-and-forth that totally throw me off. For example, given an input like `1, 3, 2, 4, 1, 5`, it’s pretty clear you could pick the subsequence `1, 3, 2, 4, 1`, and that’s an alternating set. I’m curious how long you think you could make that by adding more numbers or rearranging them!

But what really has me scratching my head is how you’d go about finding these subsequences efficiently. I’ve tried brute force methods, but they feel super inefficient and tedious for larger sequences. Has anyone implemented a smarter algorithm or found a pattern that helps?

I’ll throw in a challenge too—what if you had to do this with a very long sequence? Say tens of thousands of numbers. Would your approach change dramatically with larger datasets? I can imagine the complexity skyrocketing!

I’d love to hear your thoughts on this. What’s your strategy for tackling this kind of problem, and do you have any coding tricks up your sleeve? Let’s brainstorm some solutions together and see if we can uncover the best way to solve this intriguing puzzle!

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    2 Answers



    1. Longest Alternating Subsequence

      To tackle the problem of finding the longest alternating subsequence in a given sequence of integers, we can utilize a dynamic programming approach that operates in O(n) time complexity. The overarching idea is to maintain two variables: one for the length of the longest subsequence ending with an increasing trend and another for the length of the longest subsequence ending with a decreasing trend. We iterate through the sequence and, for each number, check its relationship with the last number of our subsequences—if it’s greater than, we can extend the increasing subsequence, whereas if it’s smaller, we extend the decreasing one. We can implement this as follows in Python:

      
def longest_alternating_subsequence(seq):
    if not seq:
        return 0
    
    up = [0] * len(seq)
    down = [0] * len(seq)
    up[0], down[0] = 1, 1
    
    for i in range(1, len(seq)):
        for j in range(i):
            if seq[i] > seq[j]:
                up[i] = max(up[i], down[j] + 1)
            elif seq[i] < seq[j]:
                down[i] = max(down[i], up[j] + 1)
    
    return max(max(up), max(down))
        
# Example usage:
input_sequence = [1, 3, 2, 4, 1, 5]
print(longest_alternating_subsequence(input_sequence))

      When dealing with very long sequences, such as tens of thousands of integers, it would be prudent to optimize this approach further by using only two variables instead of two arrays, thus reducing the space complexity to O(1). In such implementations, we would only need to keep track of the current lengths of the increasing and decreasing subsequences without storing them for each element, since we only require the length information from the previous steps. This change significantly enhances performance and is especially beneficial when working with large datasets, ensuring our algorithm remains efficient and swift. Engaging in this dynamic method not only allows for an elegant solution but also scales admirably with the input size.







    2. Longest Alternating Subsequence

      Finding the Longest Alternating Subsequence

      This sounds like such a fun challenge! I’m not really experienced with this kind of stuff, but here’s a simple approach I could think of. It’s not the most efficient, but it should show the basic idea.

      Algorithm Idea:

      You want to loop through the sequence and keep track of the last added number in our alternating sequence. If the current number is different in sign from the last one we added, we add it to our result!

      Here’s a basic code example in Python:

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

    # Start with the first element of the list
    longest = [nums[0]]

    for i in range(1, len(nums)):
        # Check if the current number has a different sign from the last added number
        if (longest[-1] < 0 and nums[i] >= 0) or (longest[-1] >= 0 and nums[i] < 0):
            longest.append(nums[i])

    return len(longest)

# Example usage:
sequence = [1, 3, 2, 4, 1, 5]
print("The length of the longest alternating subsequence is:", longest_alternating_subsequence(sequence))

      How it Works:

      The function starts by checking if the input list is empty. If it’s not empty, it initializes the longest subsequence with the first number. The loop goes through each number after the first one and checks if it has an opposite sign than the last number added. If it does, it gets added to our list!

      Performance Thoughts:

      For big lists, this method has to look at each number, so it runs in O(n). That should be manageable for lists that are tens of thousands long, but it would totally be worth checking for edge cases while testing!

      Extra Thoughts:

      I guess we could look into more advanced algorithms or even dynamic programming for better efficiency, but I hope this example helps a bit! Let me know if you have other ideas or tips on how to improve this!


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