Python Coding Test Course, Utilizing Time Complexity

Problem Solving Process

1. Understanding Time Complexity

To solve this problem, we must first consider the time complexity. Essentially, this problem can be solved with a time complexity of O(n^2). However, there is also a method to solve it with a time complexity of O(n log n), which can significantly improve the efficiency of the algorithm.

2. Solution Using Dynamic Programming

The dynamic programming approach to finding the longest increasing subsequence is as follows:

• Create an array dp to track the length of the LIS for each element. Initialize all values to 1 (since each element forms an increasing subsequence by itself).
• For each element, compare it with previous elements; if the current element is greater, update dp[i].
• Finally, find the maximum value in the dp array to determine the length of the LIS.

3. Code Implementation

Below is the code implemented in Python:

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

    dp = [1] * len(nums)  # Each element has a minimum length of 1
    for i in range(len(nums)):
        for j in range(i):
            if nums[i] > nums[j]:  # Increasing condition
                dp[i] = max(dp[i], dp[j] + 1)

    return max(dp)

4. Time Complexity Analysis

The time complexity of the above dynamic programming solution is O(n²) because there are two nested loops. However, there is also a method to solve it in O(n log n). In this method, binary search can be utilized to improve efficiency.

5. O(n log n) Approach

The O(n log n) approach uses binary search. A list is maintained to track the increasing sequence, and for each element, the appropriate position within that list is found:

import bisect

def length_of_lis(nums):
    lis = []
    for num in nums:
        pos = bisect.bisect_left(lis, num)  # Find insert position using binary search
        if pos == len(lis):
            lis.append(num)  # Add new maximum value
        else:
            lis[pos] = num  # Replace existing value
    return len(lis)

6. Running Example and Checking Results

Check if the function works properly with an example like the following:

nums = [10, 9, 2, 5, 3, 7, 101, 18]
print(length_of_lis(nums))  # Output: 4

7. Time Complexity Analysis

In the O(n log n) method, bisect.bisect_left operates in logarithmic time, so the overall time complexity is O(n log n). This provides a fast response even for large inputs, making it useful in actual coding tests.

Conclusion

This article covers the method of solving algorithm problems using Python and the concept of time complexity. Initially, the method to solve the LIS problem using dynamic programming in O(n²) was introduced, followed by improvements in efficiency using the O(n log n) method. Understanding and applying time complexity is crucial in algorithm design. I hope you continue to solve various problems and practice considering time complexity!