446. Arithmetic Slices II - Subsequence

📋 Đề Bài

Given an integer array nums, return the number of all the arithmetic subsequences of nums.

A sequence of numbers is called arithmetic if it consists of at least three elements and if the difference between any two consecutive elements is the same.

For example, [1, 3, 5, 7, 9], [7, 7, 7, 7], and [3, -1, -5, -9] are arithmetic sequences.
For example, [1, 1, 2, 5, 7] is not an arithmetic sequence.

A subsequence of an array is a sequence that can be formed by removing some elements (possibly none) of the array.

For example, [2,5,10] is a subsequence of [1,2,1,2,4,1,5,10].

The test cases are generated so that the answer fits in 32-bit integer.

Example 1:

Input: nums = [2,4,6,8,10]
Output: 7
Explanation: All arithmetic subsequence slices are:
[2,4,6]
[4,6,8]
[6,8,10]
[2,4,6,8]
[4,6,8,10]
[2,4,6,8,10]
[2,6,10]

Example 2:

Input: nums = [7,7,7,7,7]
Output: 16
Explanation: Any subsequence of this array is arithmetic.

Constraints:

1 <= nums.length <= 1000
-2³¹ <= nums[i] <= 2³¹ - 1

🧠 Thuật Toán & Kỹ Thuật

Dynamic Programming (Quy hoạch động)Hash Table (Bảng băm)Bit Manipulation (Thao tác bit)

⏱️ Thời gian O(n×m)

💾 Không gian O(n×m)

💻 Lời Giải

C++ 0446-arithmetic-slices-ii-subsequence.cpp

class Solution {
public:
    int numberOfArithmeticSlices(vector<int>& nums) {
        const int n = nums.size();
        int ans = 0;
        using ll = long long;
        vector<unordered_map<ll, int>> dp(n);
        for (int i = 1; i < n; ++i) {
            for (int j = 0; j < i; ++j) {
                ll diff = (ll)nums[i] - nums[j];
                if (INT_MAX < diff || INT_MIN > diff) {
                    continue;
                }
                int cnt = dp[j].count(diff) ? dp[j][diff] : 0;
                dp[i][diff] += cnt + 1;
                ans += cnt;
            }
        }
        return ans;
    }
};