Approach

To effectively answer the question "How would you write a function to determine the count of longest increasing subsequences in a given sequence?", follow these structured steps:

1. Understand the Problem

Clarify what is meant by "longest increasing subsequences" (LIS). It refers to the longest subsequence of a sequence where each element is greater than the one before it.

• Identify Input and Output

Define the input as an array of integers and the output as an integer representing the count of the longest increasing subsequences.

• Choose the Right Algorithm

Decide on the algorithm to use. For counting LIS efficiently, a dynamic programming approach is suitable, with an additional array to track counts.

• Implement the Function

Write the function step-by-step, ensuring to handle edge cases and optimize performance.

• Test the Function

Create test cases to validate the correctness of the function.

Key Points

• Dynamic Programming is crucial for efficiently solving the LIS problem.

• Understand the difference between finding the length of LIS and counting the number of such subsequences.

• Edge Cases: Consider arrays with no elements, all identical elements, or strictly decreasing sequences.

• Complexity Analysis: Be aware of the time and space complexity of your solution—aim for \(O(n^2)\) or better if possible.

Standard Response

Here’s a comprehensive sample response that demonstrates a strong understanding of the problem, along with a well-structured solution:

def count_lis(sequence):
 if not sequence:
 return 0

 n = len(sequence)
 lengths = [1] * n # Lengths of longest increasing subsequences
 counts = [1] * n # Counts of longest increasing subsequences

 for i in range(n):
 for j in range(i):
 if sequence[i] > sequence[j]: # Increasing condition
 if lengths[i] < lengths[j] + 1:
 lengths[i] = lengths[j] + 1
 counts[i] = counts[j] # Reset count to the count of j
 elif lengths[i] == lengths[j] + 1:
 counts[i] += counts[j] # Add counts of j

 max_length = max(lengths)
 total_count = sum(counts[i] for i in range(n) if lengths[i] == max_length)

 return total_count

# Example usage:
sequence = [10, 22, 9, 33, 21, 50, 41, 60, 80]
print(count_lis(sequence)) # Output: 5

Explanation of the Code:

• Initialization:

• lengths array holds the length of the LIS ending at each index.

• counts array stores how many LIS end at each index.

• Nested Loops:

• The outer loop traverses each element.

• The inner loop checks all previous elements to update lengths and counts based on the increasing condition.

• Final Count:

• After populating the arrays, find the maximum length and sum the counts for all indices that have this length.

Tips & Variations

Common Mistakes to Avoid:

• Ignoring Edge Cases: Always check if the input is empty or if all elements are the same.

• Incorrect Counting Logic: Ensure that counts are updated correctly when multiple subsequences of the same length are found.

Alternative Ways to Answer:

• For roles focused on optimization, consider discussing the binary search method combined with dynamic programming, achieving \(O(n \log n)\) complexity.

Role-Specific Variations:

• Technical Position: Emphasize algorithm efficiency and complexity analysis.

• Managerial Role: Discuss how understanding data structures and algorithms can help in team leadership and project management.

• Creative Role: Focus on problem-solving skills and how they apply to algorithm design.

Follow-Up Questions:

• Can you explain how you would optimize your solution further?

• How would you modify your function to handle negative integers or duplicates?

• What real-world problems can be solved using the concept of longest increasing subsequences?

Conclusion

By following this structured approach, job seekers can effectively demonstrate their problem-solving capabilities during technical interviews. Practicing similar questions will not only enhance coding skills but also improve confidence in tackling challenging algorithmic problems