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.
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.length1 <= n <= 105-105 <= differences[i] <= 105-105 <= lower <= upper <= 105This approach uses the concept of a prefix sum to efficiently monitor the cumulative effects of the differences array. By calculating the prefix sum, we can track how the potential hidden sequence evolves from a starting point. Subsequently, determining the feasible range for the starting point of the hidden sequence allows us to count how many valid hidden sequences exist within the given bounds.
In this C solution, we iterate through the differences array to calculate the prefix sum. This allows us to determine how much the hidden sequence could diverge from its initial value. By tracking the minimum and maximum prefix sums, we can calculate the minimum and maximum initial values of the hidden sequence that would fit within the specified bounds. The difference between these two bounds plus one gives us the count of valid hidden sequences.
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Time Complexity: O(n), where n is the number of elements in the differences array. Space Complexity: O(1), constant space used for tracking prefix sums.
An alternative method involves dynamically computing the range of possible values for each element in the hidden sequence. Starting with a base value, each element's value is recursively determined based on previous elements and the differences array. Finally, the overall range of valid starting points is deduced by verifying constraints imposed by lower and upper bounds.
This C code uses an array to track cumulative values derived from the differences array by incrementally adjusting the base value. Upon determining the minimum and maximum values in the range, it then calculates how many starting values fit the specified bounds, optimized via selective tracking and array manipulation.
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Java
Python
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Time Complexity: O(n), resulting from the need to iterate through the differences twice (one for accumulation and one for range computation). Space Complexity: O(n), owing to the use of additional storage for cumulative values.
| Approach | Complexity |
|---|---|
| Prefix Sum and Range Validation | Time Complexity: O(n), where n is the number of elements in the differences array. Space Complexity: O(1), constant space used for tracking prefix sums. |
| Dynamic Range Computation | Time Complexity: O(n), resulting from the need to iterate through the differences twice (one for accumulation and one for range computation). Space Complexity: O(n), owing to the use of additional storage for cumulative values. |
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