# 1027. Longest Arithmetic Sequence

Given an array A of integers, return the length of the longest arithmetic subsequence in A.

Recall that a subsequence of A is a list A[i_1], A[i_2], ..., A[i_k] with 0 <= i_1 < i_2 < ... < i_k <= A.length - 1, and that a sequence B is arithmetic if B[i+1] - B[i] are all the same value (for 0 <= i < B.length - 1).

Example 1:

Input: [3,6,9,12]
Output: 4
Explanation: 
The whole array is an arithmetic sequence with steps of length = 3.

Example 2:

Input: [9,4,7,2,10]
Output: 3
Explanation: 
The longest arithmetic subsequence is [4,7,10].

Example 3:

Input: [20,1,15,3,10,5,8]
Output: 4
Explanation: 
The longest arithmetic subsequence is [20,15,10,5].

# Solution

Approach 1: O(N^3) -- for each pair of numbers, look into the rest of the array for other items.

Approach 2: O(N^2) -- for each element in the array, store longest arithmetic sequence so far for each diff value.

# Code (Python)

Approach 1:

    def longestArithSeqLength(self, nums: List[int]) -> int:
        # O(N^3) TLE
        longest = 2
        for i, num in enumerate(nums):
            for j in range(i + 1, len(nums)):
                current_longest = 2
                diff = num - nums[j]
                target = nums[j] - diff
                for k in range(j + 1, len(nums)):
                    if nums[k] == target:
                        current_longest += 1
                        target = nums[k] - diff
                longest = max(longest, current_longest)
        return longest

Approach 2:

    def longestArithSeqLength(self, nums):
        # O(N^2)
        longest = 2
        diff_and_longest = [{0: 1}] + [{} for _ in range(len(nums) - 1)]
        for i, num in enumerate(nums):
            if i == 0:
                continue
            for j in range(i):
                diff = num - nums[j]
                diff_and_longest[i][diff] = diff_and_longest[j][diff] + 1 if diff in diff_and_longest[j] else 2
                longest = max(longest, diff_and_longest[i][diff])
        return longest

# Code (C++)

Approach 1:

Approach 2: