Problem Description
Given an integer array arr and an integer difference, return the length of the longest subsequence in arr which is an arithmetic sequence such that the difference between adjacent elements in the subsequence equals difference. A subsequence is a sequence that can be derived from arr by deleting some or no elements without changing the order of the remaining elements.
Key Insights
- The problem requires identifying subsequences where the difference between consecutive elements is constant and equals the given
difference. - A dynamic programming approach can be applied, where we maintain a record of the length of the longest arithmetic subsequence ending at each element.
- A hash table (or dictionary) can be leveraged to track the maximum length of subsequences that can be formed by considering previous elements.
Space and Time Complexity
Time Complexity: O(n), where n is the length of the input array arr. We traverse the array once.
Space Complexity: O(n), for storing the lengths of the subsequences in a hash table.
Solution
We will use a hash table to store the lengths of the longest arithmetic subsequences that can be formed ending at each element. For each element in the array, we will check if the previous element that can form an arithmetic sequence exists in our hash table. If it exists, we update our current element's maximum length based on that. If not, we set it to 1 (the element itself). This approach ensures that we only traverse the array once, making it efficient.