413 Arithmetic Slices

413. Arithmetic Slices

1. Question

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.

2. Implementation

(1) DP

思路: dp[i]表示在i位置上, 长度为3的Arithmetic Slices的个数，由于结果要求的是所有可能长度的Arithmetic Slices，我们用res累加每个位置上的dp[i]的值

class Solution {
    public int numberOfArithmeticSlices(int[] A) {
        if (A.length <= 2) {
            return 0;
        }

        int n = A.length;
        int[] dp = new int[n];
        int res = 0;

        for (int i = 2; i < n; i++) {
            if (A[i] - A[i - 1] == A[i - 1] - A[i - 2]) {
                dp[i] = 1 + dp[i - 1];
                res += dp[i];
            }
        }
        return res;
    }
}

3. Time & Space Complexity

DP: 时间复杂度O(n), 空间复杂度O(n)