## Solution

#### Approach 1: Linear Scan

Intuition

As in Smallest Range I, smaller A[i] will choose to increase their value ("go up"), and bigger A[i] will decrease their value ("go down").

Algorithm

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 + 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.

Complexity Analysis

• Time Complexity: , where is the length of the A.

• Space Complexity: , plus the space used by the builtin sorting algorithm.

Analysis written by: @awice.