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This approach involves using a 2D dynamic programming array to keep track of the lengths of arithmetic subsequences ending at different indices with various common differences. The outer loop iterates over end indices while the inner loop considers all pairs of start and end indices to update the subsequence lengths.
Time Complexity: O(n^2)
Space Complexity: O(n^2), where n is the size of the input array.
1#include <iostream>
2#include <vector>
3#include <unordered_map>
4using namespace std;
5
6int longestArithSeqLength(vector<int>& nums) {
7 if (nums.size() <= 2) return nums.size();
8 vector<unordered_map<int, int>> dp(nums.size());
9 int maxLength = 2;
10 for (int i = 0; i < nums.size(); i++) {
11 for (int j = 0; j < i; j++) {
12 int diff = nums[i] - nums[j];
13 dp[i][diff] = dp[j].count(diff) > 0 ? dp[j][diff] + 1 : 2;
14 maxLength = max(maxLength, dp[i][diff]);
15 }
16 }
17 return maxLength;
18}
19
20int main() {
21 vector<int> nums = {3, 6, 9, 12};
22 cout << longestArithSeqLength(nums) << endl;
23 return 0;
24}This C++ solution uses a vector of hashmaps to store the length of arithmetic subsequences by their common differences. For each pair `(j, i)`, we calculate the difference and check if it already exists in `dp[j]`. If it does, we can extend this sequence to include `nums[i]`.
This approach uses a simplified dynamic programming technique with HashMaps and tracks differences and their counts for each number. By leveraging HashMap structures, we manage subsequence lengths more efficiently and avoid using unnecessary space for non-existent differences.
Time Complexity: O(n^2)
Space Complexity: O(n^2) in the worst case but often much less due to sparse differences.
1using System.Collections.Generic;
public class Solution {
public int LongestArithSeqLength(int[] nums) {
if (nums.Length <= 2) return nums.Length;
var dp = new Dictionary<int, int>[nums.Length];
int maxLength = 2;
for (int i = 0; i < nums.Length; i++) {
dp[i] = new Dictionary<int, int>();
for (int j = 0; j < i; j++) {
int diff = nums[i] - nums[j];
if (dp[j].ContainsKey(diff)) {
dp[i][diff] = dp[j][diff] + 1;
} else {
dp[i][diff] = 2;
}
maxLength = Math.Max(maxLength, dp[i][diff]);
}
}
return maxLength;
}
public static void Main() {
int[] nums = {9, 4, 7, 2, 10};
var sol = new Solution();
Console.WriteLine(sol.LongestArithSeqLength(nums));
}
}This C# code handles the longest arithmetic subsequence problem by establishing dictionaries for each element in the array, keeping score of different arithmetic sequences iteratively and effectively.