#2145

Medium

LeetCode Problem Workspace

Count the Hidden Sequences

Given a sequence of differences, count how many hidden sequences fit within a range using an array and prefix sum approach.

array prefix sum

Problem Statement

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

Examples

Example 1

Input: differences = [1,-3,4], lower = 1, upper = 6

Output: 2

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

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

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 Approach

Prefix Sum Calculation

To solve this problem, you can fix the first element of the hidden sequence to any value x. Then, calculate the rest of the sequence using prefix sums of the given differences. This approach allows you to iterate through the differences array and compute all subsequent values based on the first value of the hidden sequence.

Range Filtering

After calculating the hidden sequence, you need to ensure that each value in the sequence lies within the inclusive range defined by lower and upper. If the first value x falls outside the range, discard that particular hidden sequence, and only keep the valid sequences where all elements stay within the specified bounds.

Optimized Calculation

To efficiently solve the problem, consider the time complexity and avoid unnecessary recalculations. Iterate through the differences array once, adjusting for the first value, and compute subsequent values in constant time. This approach will guarantee the solution works within O(n) time.

Complexity Analysis

Metric	Value
Time	O(n)
Space	O(1)

The time complexity of the solution is O(n) since we only need to iterate through the differences array once to compute the hidden sequence. The space complexity is O(1) because we do not need any additional data structures beyond a few variables to track the first value and sequence elements.

What Interviewers Usually Probe

Understanding prefix sum calculation is crucial to solving this problem.
Candidates should handle range filtering effectively to ensure correct sequences.
Watch for solutions that improperly iterate or recompute values multiple times.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly compute the subsequent sequence values based on the first fixed value.
Not accounting for sequences where any value exceeds the specified range.
Using an inefficient approach that leads to higher time complexity than O(n).

Follow-up variants

Consider extending this problem to support different types of sequences, such as geometric sequences.
What if the sequence is not constrained by a fixed range? How would that change the solution?
Modify the problem to account for sequences with additional constraints, such as specific values at certain indices.

FAQ

What is the approach for solving the 'Count the Hidden Sequences' problem?

The approach involves fixing the first element of the hidden sequence and using prefix sums to calculate the rest of the sequence while ensuring all values lie within the specified range.

How do I handle sequences that exceed the specified range?

You need to filter out sequences where any value exceeds the given range [lower, upper] to ensure the sequence is valid.

What is the time complexity of the solution?

The time complexity is O(n), where n is the length of the differences array, since we only iterate through it once.

Can the 'Count the Hidden Sequences' problem be solved without prefix sums?

While it’s possible to compute the hidden sequence without using prefix sums, prefix sums make the calculation more efficient by allowing you to compute all subsequent sequence values in constant time.

What happens if the first value of the sequence is out of bounds?

If the first value of the sequence falls outside the range [lower, upper], that sequence is not valid, and you should discard it.

terminal

Solution

Solution 1: Prefix Sum

Since the array $\textit{differences}$ is already determined, the difference between the maximum and minimum values of the elements in the array $\textit{hidden}$ is also fixed. We just need to ensure that this difference does not exceed $\textit{upper} - \textit{lower}$.

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class Solution:
    def numberOfArrays(self, differences: List[int], lower: int, upper: int) -> int:
        x = mi = mx = 0
        for d in differences:
            x += d
            mi = min(mi, x)
            mx = max(mx, x)
        return max(upper - lower - (mx - mi) + 1, 0)

Continue Practicing

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