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.

 

Approach #1: 

class Solution {
public:
    int numberOfArithmeticSlices(vector<int>& A) {
        int size = A.size();
        int ans = 0;
        if (size < 3) return ans;
        vector<int> B(size-1, 0);
        
        for (int i = 1; i < size; ++i) 
            B[i-1] = A[i] - A[i-1];
        
        for (int i = 0; i < B.size(); ) {
            int j = i;
            while (j + 1 < B.size() && B[i] == B[j+1]) ++j;
            int k = j - i + 1;
            if (k >= 2) ans += k * (k - 1) / 2;
            i = j + 1;
        }
        
        return ans;
    }
};

　　

Approach #2: DP. 

public class Solution {
    public int numberOfArithmeticSlices(int[] A) {
        int[] dp = new int[A.length];
        int sum = 0;
        for (int i = 2; i < dp.length; i++) {
            if (A[i] - A[i - 1] == A[i - 1] - A[i - 2]) {
                dp[i] = 1 + dp[i - 1];
                sum += dp[i];
            }
        }
        return sum;
    }
}

　　

Approach #3: constant space DP. 

public class Solution {
    public int numberOfArithmeticSlices(int[] A) {
        int dp = 0;
        int sum = 0;
        for (int i = 2; i < A.length; i++) {
            if (A[i] - A[i - 1] == A[i - 1] - A[i - 2]) {
                dp = 1 + dp;
                sum += dp;
            } else
                dp = 0;
        }
        return sum;
    }
}

　　

Approach #4: Formula.

public class Solution {
    public int numberOfArithmeticSlices(int[] A) {
        int count = 0;
        int sum = 0;
        for (int i = 2; i < A.length; i++) {
            if (A[i] - A[i - 1] == A[i - 1] - A[i - 2]) {
                count++;
            } else {
                sum += (count + 1) * (count) / 2;
                count = 0;
            }
        }
        return sum += count * (count + 1) / 2;
    }
}

　　

Analysis:

https://leetcode.com/problems/arithmetic-slices/solution/