Count the Hidden Sequences

Count the Hidden Sequences

What is this problem about?

The Count the Hidden Sequences coding problem provides you with an array of differences between successive elements of an unknown "hidden" sequence. You are also given a lower and upper boundary. Your task is to determine how many possible sequences exist such that every element in the sequence remains within the inclusive range $[lower, upper]$.

For example, if the differences are $[1, -3, 4]$, a possible sequence could be $[0, 1, -2, 2]$. If the boundaries are $[1, 5]$, we need to find how many such sequences start with a value that keeps all subsequent values between 1 and 5.

Why is this asked in interviews?

Companies like Meta, Amazon, and TikTok ask this to see if a candidate can handle relative constraints. It tests the ability to recognize that the entire sequence's range is fixed by its differences, and the only "freedom" we have is choosing the starting element. It evaluates proficiency with the prefix sum interview pattern and the ability to find the range (min and max) of a derived set of values.

Algorithmic pattern used

This problem relies on Prefix Sums and Range Analysis.

1. Assume the sequence starts at 0.
2. Calculate all subsequent elements by accumulating the differences (prefix sums).
3. Find the maximum value (max_val) and the minimum value (min_val) in this assumed sequence.
4. The total spread of the sequence is max_val - min_val.
5. If this spread is greater than the allowed range upper - lower, then no such sequence is possible.
6. Otherwise, the number of possible starting positions is (upper - lower) - spread + 1.

Example explanation

Differences: [1, 2], Boundaries: [3, 10]

1. Start at 0: Sequence is [0, 1, 3].
2. min_val = 0, max_val = 3.
3. Spread = $3 - 0 = 3$.
4. Allowed range = $10 - 3 = 7$.
5. Number of starting positions: $7 - 3 + 1 = 5$. Possible sequences start at 3, 4, 5, 6, 7. (e.g., if we start at 7: [7, 8, 10] — all within $[3, 10]$).

Common mistakes candidates make

• Trying to simulate every start value: This is too slow if the boundary range is large.
• Off-by-one errors: Forgetting to add 1 when calculating the number of valid starting values.
• Overflow: Not using long integers for the prefix sums, as differences can accumulate to exceed standard integer limits.

Interview preparation tip

When a problem involves "hidden" values defined by differences, always simplify by assuming a start value of 0. This reveals the "shape" of the data, allowing you to treat the problem as a simple geometry or range-matching exercise.