Problem Description
You are given an integer array nums and a positive integer k. A subsequence sub of nums with length x is called valid if it satisfies: (sub[0] + sub[1]) % k == (sub[1] + sub[2]) % k == ... == (sub[x - 2] + sub[x - 1]) % k. Return the length of the longest valid subsequence of nums.
Key Insights
- A valid subsequence must maintain a consistent modulo k value for the sum of every adjacent pair of elements.
- The problem can be approached using dynamic programming to track the maximum length of subsequences that meet the validity condition.
- Careful handling of indices and modulo operations is essential to ensure that the subsequence remains valid as new elements are added.
Space and Time Complexity
Time Complexity: O(n^2), where n is the length of the nums array, due to the nested iteration to check pairs. Space Complexity: O(n), for storing the lengths of valid subsequences.
Solution
To solve the problem, we can use a dynamic programming approach. We will create an array dp where dp[i] represents the length of the longest valid subsequence that ends with the element at index i. For each pair of indices (i, j) where i < j, we will check if (nums[i] + nums[j]) % k is the same as the previous pairs. If they are valid, we update dp[j] to be the maximum of its current value and dp[i] + 1. Finally, we return the maximum value in the dp array, which represents the length of the longest valid subsequence.