Problem Description
Given an array nums of integers, return the length of the longest arithmetic subsequence in nums. A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements. A sequence seq is arithmetic if seq[i + 1] - seq[i] are all the same value.
Key Insights
- An arithmetic subsequence has a constant difference between consecutive elements.
- The problem can be solved using a dynamic programming approach to keep track of the lengths of arithmetic subsequences with specific differences.
- A hash table (dictionary) can be used to map each difference to the length of the longest subsequence found so far for that difference.
Space and Time Complexity
Time Complexity: O(n^2), where n is the length of the input array.
Space Complexity: O(n * d), where d is the range of possible differences.
Solution
To solve the problem, we will use a dynamic programming approach. We will iterate through each pair of indices in the array and calculate the difference between the elements. We will maintain a dictionary (hash table) that keeps track of the lengths of the arithmetic subsequences for each difference encountered. For each pair of indices (i, j), we will update our dictionary based on the difference and the previously calculated lengths. The maximum length found in the dictionary will be our answer.