## DEV Community

Abhishek Chaudhary

Posted on

# Count the Hidden Sequences

You are given a 0-indexed array of `n` integers `differences`, which describes the differences between each pair of consecutive integers of a hidden sequence of length `(n + 1)`. More formally, call the hidden sequence `hidden`, then we have that `differences[i] = hidden[i + 1] - hidden[i]`.

You are further given two integers `lower` and `upper` that describe the inclusive range of values `[lower, upper]` that the hidden sequence can contain.

• For example, given `differences = [1, -3, 4]`, `lower = 1`, `upper = 6`, the hidden sequence is a sequence of length `4` whose elements are in between `1` and `6` (inclusive).
• `[3, 4, 1, 5]` and `[4, 5, 2, 6]` are possible hidden sequences.
• `[5, 6, 3, 7]` is not possible since it contains an element greater than `6`.
• `[1, 2, 3, 4]` is not possible since the differences are not correct.

Return the number of possible hidden sequences there are. If there are no possible sequences, return `0`.

Example 1:

Input: differences = [1,-3,4], lower = 1, upper = 6
Output: 2
Explanation: The possible hidden sequences are:

• [3, 4, 1, 5]
• [4, 5, 2, 6] Thus, we return 2.

Example 2:

Input: differences = [3,-4,5,1,-2], lower = -4, upper = 5
Output: 4
Explanation: The possible hidden sequences are:

• [-3, 0, -4, 1, 2, 0]
• [-2, 1, -3, 2, 3, 1]
• [-1, 2, -2, 3, 4, 2]
• [0, 3, -1, 4, 5, 3] Thus, we return 4.

Example 3:

Input: differences = [4,-7,2], lower = 3, upper = 6
Output: 0
Explanation: There are no possible hidden sequences. Thus, we return 0.

Constraints:

• `n == differences.length`
• `1 <= n <= 105`
• `-105 <= differences[i] <= 105`
• `-105 <= lower <= upper <= 105`

SOLUTION:

``````class Solution:
def numberOfArrays(self, differences: List[int], lower: int, upper: int) -> int:
pos = [0]
for d in differences:
pos.append(pos[-1] + d)
least = min(pos)
pos = [p + lower - least for p in pos]
return max(upper - max(pos) + 1, 0)
``````