413. Arithmetic Slices

A sequence of number 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, these are arithmetic sequence:

1, 3, 5, 7, 9
7, 7, 7, 7
3, -1, -5, -9

The following sequence is not arithmetic.

1, 1, 2, 5, 7

A zero-indexed array A consisting of N numbers is given. A slice of that array is any pair of integers (P, Q) such that 0 <= P < Q < N.

A slice (P, Q) of array A is called arithmetic if the sequence: A[P], A[p + 1], ..., A[Q - 1], A[Q] is arithmetic. In particular, this means that P + 1 < Q.

The function should return the number of arithmetic slices in the array A.

Example:

A = [1, 2, 3, 4]

return: 3, for 3 arithmetic slices in A: [1, 2, 3], [2, 3, 4] and [1, 2, 3, 4] itself.

dp[i]代表以當前這個數結尾的等差數列數量。 dp[i] = dp[i - 1] + 1

// Dynamic Programming
int numberOfArithmeticSlices(vector<int>& A) { // time: O(n); space: O(n)
    if (A.empty()) return 0;
    int n = A.size(), res = 0;
    vector<int> dp(n, 0); // the number of arithmetic slices ending with A[i]
    for (int i = 2; i < n; ++i) {
        if (A[i] - A[i - 1] == A[i - 1] - A[i - 2]) {
            dp[i] = dp[i - 1] + 1;
            res += dp[i];
        }
    }
    return res;
}

因為dp[i]只跟dp[i - 1]有關，所以可以簡化空間使用到一個變數就好。cur代表以當前這個數字結尾的等差數列數量。

// Space Optimized Dynamic Programming
int numberOfArithmeticSlices(vector<int>& A) { // time: O(n); space: O(1)
    if (A.empty()) return 0;
    int res = 0, cur = 0; // cur: the number of arithmetic slices ending with A[i]
    for (int i = 2; i < A.size(); ++i) {
        if (A[i] - A[i - 1] == A[i - 1] - A[i - 2]) {
            ++cur;
            res += cur;
        } else {
            cur = 0;
        }
    }
    return res;
}