Find the Maximum Length of Valid Subsequence II

Problem Description

Ah, the classic “Find the Maximum Length of Valid Subsequence II” problem! It’s like trying to find the longest line at a coffee shop, but instead of caffeine, you’re just trying to figure out how to maximize a sequence of numbers. Imagine you’re at a party, and you want to dance with the longest line of friends without stepping on anyone’s toes. But wait! There’s a catch: you can only dance with friends whose dance moves (numbers) fit a certain pattern.

In this problem, you’re given an array of integers and a number k. Your mission, should you choose to accept it, is to find the maximum length of a valid subsequence where the last number mod k equals the first number mod k. Sounds simple, right? Well, it’s like trying to find a matching pair of socks in a laundry basket—easy in theory, but a real challenge in practice!

Code Solution


class Solution:
    def maximumLength(self, nums: list[int], k: int) -> int:
        dp = [[0] * k for _ in range(k)]
        for x in nums:
            for y in range(k):
                dp[x % k][y] = dp[y][x % k] + 1
        return max(map(max, dp))

Approach Explanation

The approach here is as elegant as a swan gliding across a lake. We use a dynamic programming table dp where dp[i][j] represents the maximum length of a valid subsequence ending with a number that gives a remainder of i when divided by k, and the next number should give a remainder of j.

We iterate through each number in the input list, updating our dp table based on the current number’s remainder. It’s like building a tower of blocks, where each block represents a valid subsequence, and we’re just trying to stack them as high as possible without them toppling over.

Time and Space Complexity

Complexity Type	Complexity
Time Complexity	O(n * k²)
Space Complexity	O(k²)

Real-World Example

Imagine you’re organizing a relay race, and each runner has a specific number of laps they can run. You want to maximize the number of laps run by ensuring that the last runner’s lap count matches the first runner’s lap count modulo some number k. This problem is akin to ensuring that the baton is passed smoothly without any hiccups, maximizing the total distance covered by the team.