Longest arithmetic sequence

Time: O(N^2); Space: O(N^2); medium

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: A = [3,6,9,12]

Output: 4

Explanation:

  • The whole array is an arithmetic sequence with steps of length = 3.

Example 2:

Input: A = [9,4,7,2,10]

Output: 3

Explanation:

  • The longest arithmetic subsequence is [4,7,10].

Example 3:

Input: A = [20,1,15,3,10,5,8]

Output: 4

Explanation:

  • The longest arithmetic subsequence is [20,15,10,5].

Constraints:

  • 2 <= len(A) <= 2000

  • 0 <= A[i] <= 10000

1. Dynamic programming [O(N^2), O(N^2)]

[1]:

import collections

class Solution1(object):
    """
    Time: O(N^2)
    Space: O(N^2)
    """
    def longestArithSeqLength(self, A):
        """
        :type A: List[int]
        :rtype: int
        """
        dp = collections.defaultdict(int)

        for i in range(len(A)-1):
            for j in range(i+1, len(A)):
                v =  A[j]-A[i]
                dp[v, j] = max(dp[v, j], dp[v, i] + 1)

        return max(dp.values()) + 1

[2]:

s = Solution1()

A = [3,6,9,12]
assert s.longestArithSeqLength(A) == 4

A = [9,4,7,2,10]
assert s.longestArithSeqLength(A) == 3

A = [20,1,15,3,10,5,8]
assert s.longestArithSeqLength(A) == 4