Count the Hidden Sequences

Problem Description

You are given a 0-indexed array of integers differences, which describes the differences between each pair of consecutive integers of a hidden sequence of length (n + 1). You also have two integers lower and upper that describe the inclusive range of values [lower, upper] that the hidden sequence can contain. The task is to return the number of possible hidden sequences.

Key Insights

• The hidden sequence can be constructed using the cumulative sums of the differences array.
• The first element of the hidden sequence can be adjusted within the bounds defined by lower and upper.
• The minimum and maximum values of the hidden sequence can be calculated based on the cumulative sums and the starting element.
• The number of valid starting points can be derived from the calculated range of the hidden sequence.

Space and Time Complexity

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

Solution

To solve this problem, we will:

1. Calculate the cumulative sum of the differences array, which will help in determining the range of values in the hidden sequence.
2. Identify the minimum and maximum values that can be formed with the cumulative sums.
3. Derive the range of valid starting points for the hidden sequence based on the constraints provided by lower and upper.
4. Count the number of valid starting points by checking the overlap of the calculated range with the range defined by lower and upper.

Code Solutions