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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 Explanation for Count the Hidden Sequences

This problem involves finding the number of possible hidden sequences given differences between consecutive elements, and upper and lower bounds for the sequence's elements.

Core Idea: The key observation is that the differences between consecutive elements determine the relative values within the hidden sequence. The absolute values of the hidden sequence are determined by the first element, which can vary within a certain range.

Algorithm:

  1. Prefix Sum: We calculate the prefix sum of the differences array. This prefix sum, at each index i, represents the difference between the i+1-th element and the first element of the hidden sequence.

  2. Minimum and Maximum: While calculating the prefix sum, we also track the minimum (mi) and maximum (mx) prefix sums encountered. The difference between the maximum and minimum prefix sum represents the range of possible values spanned by the elements of the hidden sequence relative to the first element.

  3. Range Check: The total range spanned by the hidden sequence relative to its first element is mx - mi. We can shift the entire sequence up or down to fit within the provided [lower, upper] range. This is possible if and only if mx - mi <= upper - lower. If it's not possible, we return 0.

  4. Count Valid Sequences: If the condition in step 3 holds, the number of possible hidden sequences is determined by how much "wiggle room" we have within the [lower, upper] range. This wiggle room is given by upper - lower - (mx - mi) + 1.

Time and Space Complexity:

  • Time Complexity: O(n), where n is the length of the differences array. This is because we iterate through the differences array once to calculate the prefix sum, minimum, and maximum values.

  • Space Complexity: O(1). We only use a constant amount of extra space to store variables like x, mi, mx.

Code Implementation (Python):

class Solution:
    def numberOfArrays(self, differences: List[int], lower: int, upper: int) -> int:
        x = mi = mx = 0  # Initialize prefix sum, min, and max
        for d in differences:
            x += d
            mi = min(mi, x)  # Update minimum prefix sum
            mx = max(mx, x)  # Update maximum prefix sum
 
        # Calculate the range of possible first elements
        range_needed = mx - mi
 
        # Check if a valid sequence is possible and calculate the count
        if upper - lower < range_needed:
            return 0  # No valid sequences
        else:
            return upper - lower - range_needed + 1  # Number of valid sequences

The implementations in other languages (Java, C++, Go, TypeScript) follow the same logic and have the same time and space complexity. They simply use the appropriate language's syntax for prefix sum calculation, min/max tracking, and range checks.