Problem Informations

• Problem Index: 2145
• Problem Link: Count the Hidden Sequences
• Topics: Array, Prefix Sum

Problem Description

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 $\text{differences}[i] = \text{hidden}[i + 1] - \text{hidden}[i]$.

You are further given two integers lower and upper that describe the inclusive range of values $[ \text{lower}, \text{upper} ]$ 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:

• $n == \text{differences.length}$
• $1 \leq n \leq 10^5$
• $-10^5 \leq \text{differences}[i] \leq 10^5$
• $-10^5 \leq \text{lower} \leq \text{upper} \leq 10^5$

Intuition

The problem can be understood as reconstructing a sequence of numbers based on the cumulative sum of given differences between consecutive elements. The task is to find how many such sequences are possible within a defined numeric range. Intuitively, the approach is to determine the range of valid starting values for the hidden sequence, such that when the differences are cumulatively applied, the entire sequence remains within the defined bounds.

Approach

The approach to solve this problem involves simulating the process of sequence reconstruction using the provided differences to establish the potential range of the hidden sequence. Here is a step-by-step explanation:

1. Initialize Variables: We start by setting current_value, max_value, and min_value to 0. These will be used to track the progression of the cumulative sum as we iterate through the differences array.

2. Iterate and Compute Cumulative Sums: As we loop over each difference in the differences array, we update current_value by adding each difference. This cumulative sum simulates the transformation from one element in the hidden sequence to the next.

3. Track Extremes: During the iteration, we continuously update max_value to be the maximum of itself and current_value, and similarly, min_value to be the minimum between itself and current_value. This allows us to capture the maximum positive deviation and maximum negative deviation from the initial starting point.

4. Calculate Possible Starting Points: The possible range for the hidden sequence’s initial value, hidden[0], can be derived from the constraints lower and upper. Specifically, the sequence is valid if for every value in it is within the inclusive range $[ \text{lower}, \text{upper} ]$. Therefore, determine the number of valid starting values as: $$ \text{number of valid sequences} = \max(\text{upper} - \text{lower} - \text{max\_value} + \text{min\_value} + 1, 0) $$

5. Return Result: If calculations yield a positive range, return that number as the count of valid starting points; otherwise, return 0, indicating no valid sequence is possible.

This algorithm effectively captures the range of potential shifts of the sequence within the given bounds and efficiently determines the number of valid sequences by evaluating the derived range criteria. The use of simple arithmetic and tracking of extremum values optimizes the solution to meet constraints suitable for large input sizes.

Code

C++

class Solution {
public:
    int numberOfArrays(vector<int>& differences, int lower, int upper) {
        // Initialize the current value, maximum, and minimum to track sequence bounds
        long long current_value = 0;
        long long max_value = 0;
        long long min_value = 0;

        // Calculate the running total of differences to find the bounds
        for (int difference : differences) {
            current_value += difference;
            max_value = max(max_value, current_value);
            min_value = min(min_value, current_value);
        }

        // Calculate the number of possible sequences using the derived bounds
        return max(int(upper - lower - max_value + min_value + 1), 0);
    }
};

Python

class Solution:
    def numberOfArrays(self, differences, lower, upper):
        # Initialize the current value, maximum, and minimum to track sequence bounds
        current_value = 0
        max_value = 0
        min_value = 0

        # Calculate the running total of differences to find the bounds
        for difference in differences:
            current_value += difference
            max_value = max(max_value, current_value)
            min_value = min(min_value, current_value)

        # Calculate the number of possible sequences using the derived bounds
        return max(int(upper - lower - max_value + min_value + 1), 0)

Complexity

• Time complexity: $O(n)$
  The function iterates through the differences array once, performing constant-time operations (addition, comparison) for each element. Therefore, the time complexity is linear with respect to the size of the input array, which is $n$. Thus, the time complexity is $O(n)$.
• Space complexity: $O(1)$
  The function uses a fixed amount of additional space to store variables like current_value, max_value, and min_value. These occupy a constant amount of space irrespective of the size of the input array. Thus, the space complexity is $O(1)$.