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].

Note:
2 <= A.length <= 2000
0 <= A[i] <= 10000

Find the Longest Arithmetic Sequence by Dynamic Programming Algorithm

Let dp[i][diff] be the maximum length of the Longest Arithmetic Sequence with difference diff and the sequence ends at index i, then we can do a O(N^2) loop to update the maximum length.

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class Solution {
public:
    int longestArithSeqLength(vector<int>& A) {
        if (A.empty()) return 0;
        int n = A.size();
        unordered_map<int, unordered_map<int, int>> dp;
        int ans = 0;
        for (int i = 0; i < n; ++ i) {
            for (int j = 0; j < i; ++ j) {
                int diff = A[i] - A[j];
                dp[i][diff] = max(dp[i][diff], dp[j][diff] + 1);
                ans = max(ans, dp[i][diff]);
            }
        }
        return ans + 1;
    }
};

class Solution {
public:
    int longestArithSeqLength(vector<int>& A) {
        if (A.empty()) return 0;
        int n = A.size();
        unordered_map<int, unordered_map<int, int>> dp;
        int ans = 0;
        for (int i = 0; i < n; ++ i) {
            for (int j = 0; j < i; ++ j) {
                int diff = A[i] - A[j];
                dp[i][diff] = max(dp[i][diff], dp[j][diff] + 1);
                ans = max(ans, dp[i][diff]);
            }
        }
        return ans + 1;
    }
};

The longest sequence is the maxmium value occured in dp[i][diff] where i is from 0 to n-1. We use the nested unordered_map (hash map) to store the two dimensional array with O(1) access. The default value is 0 if the key is not existent in the unordered_map.

If the difference is given, and you can find the Longest Arithmetic Sequence using DP as well: Finding Out the Longest Arithmetic Subsequence of Given Difference using Dynamic Programming Algorithm

–EOF (The Ultimate Computing & Technology Blog) —