Find the Number of Copy Arrays

Problem Description

You are given an array original of length n and a 2D array bounds of length n x 2, where bounds[i] = [u_i, v_i]. You need to find the number of possible arrays copy of length n such that:

1. (copy[i] - copy[i - 1]) == (original[i] - original[i - 1]) for 1 <= i <= n - 1.
2. u_i <= copy[i] <= v_i for 0 <= i <= n - 1.

Return the number of such arrays.

Key Insights

• The difference between consecutive elements in the copy array must match that of the original array.
• For each element in the copy array, the valid range is dictated by the given bounds.
• The problem can be approached by calculating the valid range for the first element and then propagating the constraints through the array.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we can follow these steps:

1. Calculate the difference between consecutive elements in the original array.
2. Initialize the valid range for the first element of copy based on bounds[0].
3. Iterate through the rest of the elements, updating the valid range for each copy[i] based on the previous element's range and the difference calculated from the original array.
4. For each index, determine the number of valid integers that fall within the updated range.
5. The total number of valid arrays is the product of valid counts across all indices.

Code Solutions