Longest Arithmetic Subsequence of Given Difference

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.

Example 1:

Input: arr = [1,2,3,4], difference = 1 Output: 4 Explanation: The longest arithmetic subsequence is [1,2,3,4].

Example 2:

Input: arr = [1,3,5,7], difference = 1 Output: 1 Explanation: The longest arithmetic subsequence is any single element.

Example 3:

Input: arr = [1,5,7,8,5,3,4,2,1], difference = -2 Output: 4 Explanation: The longest arithmetic subsequence is [7,5,3,1].

Constraints:

• 1 <= arr.length <= 105
• -104 <= arr[i], difference <= 104

Code

DP - TLE

Time Complexity: $O(n^2)$, Space Complexity: $O(n)$

class Solution {
public:
    int longestSubsequence(vector<int>& arr, int difference) {
        int n = arr.size();
        // vector<int> dp(n, 1);
 
        // don't use memset(dp, 1, sizeof(dp)) !!, since memset works
        // on byte-to-byte basis.
        int dp[n];
        for(int i = 0; i < n; i++) {
            dp[i] = 1;
        }
 
        int res = 1;
        for(int i = 1; i < n; i++) {
            for(int j = i - 1; j >= 0; j--) {
                if((arr[i] - arr[j]) == difference) {
                    dp[i] = max(dp[i], dp[j] + 1);
                    res = max(res, dp[i]);
                }
            }
        }
 
        return res;
    }
};

DP - with hashmap

因為 TLE，所以必須加速，要加速找到 (arr[i] - arr[j]) == difference 的方式就是使用 hashmap。

Time Complexity: $O(n)$, Space Complexity: $O(n)$

class Solution {
public:
    int longestSubsequence(vector<int>& arr, int difference) {
        unordered_map<int, int> dp;
 
        int res = 1;
        for(int i = 0; i < arr.size(); i++) {
            if(dp.find(arr[i] - difference) != dp.end()) {
                dp[arr[i]] = dp[arr[i] - difference] + 1;
                res = max(res, dp[arr[i]]);
            } else {
                dp[arr[i]] = 1;
            }
        }
 
        return res;
    }
};

Source

• Longest Arithmetic Subsequence of Given Difference - LeetCode