446. Arithmetic Slices II - Subsequence (Hard)

Description Link to heading

446. Arithmetic Slices II - Subsequence (Hard)

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

Solution Link to heading

This problem evidently calls for a dynamic programming approach. For instance, let $dp[i]$ denote the length of the subsequence ending with nums[i]. To transition from $dp[j]$, we must be aware of $nums[i] - nums[j]$. Consequently, the $dp$ array requires two dimensions: the first dimension represents the index of the array $nums$, and the second dimension denotes the difference $diff$. As $diff$ can be substantial and even negative, we employ a hash table for record-keeping.

Hence, vector<unordered_map<long, long>> dp.

Furthermore, there’s the requirement that the array length must be at least $3$. Eliminating sequences of length $2$ during the state transition process can be somewhat intricate. However, we can easily remove length-$2$ sequences after computation by subtracting $\binom{n}{2}$ from the result.

Code Link to heading

class Solution {
  public:
    int numberOfArithmeticSlices(vector<int> &nums) {
        if (nums.size() < 3) {
            return 0;
        }
        int n = nums.size();
        int res = 0;
        vector<unordered_map<long, long>> dp(n);
        for (int i = 1; i < n; ++i) {
        	for (int j = 0; j < i; ++j) {
        		long diff = (long)nums[i] - nums[j];
        		dp[i][diff] += 1 + dp[j][diff];
        		res += 1 + dp[j][diff];
        	}
        }
        return res - n * (n - 1) / 2;
    }
};