910. Smallest Range II

Given an array A of integers, for each integer A[i] we need to choose either x = -K or x = K, and add x to A[i] (only once).

After this process, we have some array B.

Return the smallest possible difference between the maximum value of B and the minimum value of B.

Example 1:

Input: A = [1], K = 0
Output: 0
Explanation: B = [1]

Example 2:

Input: A = [0,10], K = 2
Output: 6
Explanation: B = [2,8]

Example 3:

Input: A = [1,3,6], K = 3
Output: 3
Explanation: B = [4,6,3]

Note:

1. 1 <= A.length <= 10000
2. 0 <= A[i] <= 10000
3. 0 <= K <= 10000

 

Approach #1: C++.

class Solution {
public:
    int smallestRangeII(vector<int>& A, int K) {
        int N = A.size();
        sort(A.begin(), A.end());
        int ans = A[N-1] - A[0];
        
        for (int i = 0; i < A.size()-1; ++i) {
            int a = A[i], b = A[i+1];
            int high = max(A[N-1]-K, a+K);
            int low = min(A[0]+K, b-K);
            ans = min(ans, high-low);
        }
        
        return ans;
    }
};

　　

Analysis:

We can formalize the above concept: if A[i] < A[j], we don't need to consider when A[i] goes down while A[j] goes up. This is because the interval (A[i] + K, A[j] - K) is a subset of (A[i] - K, A[j] + K) (here, (a, b) for a > b denotes (b, a) instead.)

That means that it is never worse to choose (up, down) instead of (down, up). We can prove this claim that one interval is a subset of another, by showing both A[i] + K and A[j] - K are between A[i] - K and A[j] + K.

For sorted A, say A[i] is the largest i that goes up. Then A[0] + K, A[i] + K, A[i+1] - K, A[A.length - 1] - K are the only relevant values for calculating the answer: every other value is between one of these extremal values.