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Count the Hidden Sequences

Number: 2249

Difficulty: Medium

Paid? No

Companies: Amazon, TikTok

Problem Description

Given an array of differences describing the differences between consecutive elements of a hidden sequence, and given lower and upper bounds for the values in the hidden sequence, determine how many valid hidden sequences exist such that every element in the sequence is within the inclusive range [lower, upper]. The key insight is that the hidden sequence can be described as a starting value plus prefix sums of the differences.

Key Insights

  • The hidden sequence can be represented as hidden[0] = x and hidden[i] = x + prefix[i], where prefix[i] is the cumulative sum of differences up to index i-1.
  • Every element in the sequence must satisfy the constraint lower <= x + prefix[i] <= upper.
  • By finding the minimum and maximum cumulative prefix sum values, we can determine the valid range for the starting value x.
  • The answer is the count of integer x values in that range.

Space and Time Complexity

Time Complexity: O(n) where n is the length of the differences array. Space Complexity: O(1) extra space (apart from the input), as we compute the cumulative sums on the fly.

Solution

The solution involves computing the prefix sums of the differences array to determine the minimum and maximum deviation from the starting value x. Let m = minimum prefix sum and M = maximum prefix sum. To ensure that every element of the hidden sequence is within [lower, upper], the starting value x must satisfy:

  • x >= lower - m and x <= upper - M. Thus, the number of valid starting values is given by: (upper - M) - (lower - m) + 1, if this range is non-negative. If the computed range is negative, then no valid hidden sequence exists (return 0).

We then implement this logic in code, ensuring clear variable names and comments at every step.

Code Solutions

# Python solution with line-by-line comments
def count_hidden_sequences(differences, lower, upper):
    # Initialize the current cumulative sum and the tracking min and max values
    current_prefix = 0
    min_prefix = 0
    max_prefix = 0
    
    # Iterate through differences array to compute prefix sum
    for diff in differences:
        current_prefix += diff
        # Update the minimum prefix sum encountered so far
        min_prefix = min(min_prefix, current_prefix)
        # Update the maximum prefix sum encountered so far
        max_prefix = max(max_prefix, current_prefix)
    
    # Determine the valid range for the starting value x
    start_lower_bound = lower - min_prefix   # x must be at least this
    start_upper_bound = upper - max_prefix     # x must be at most this
    
    # Calculate the number of valid hidden sequences
    if start_lower_bound > start_upper_bound:
        return 0
    return start_upper_bound - start_lower_bound + 1

# Example usage:
print(count_hidden_sequences([1, -3, 4], 1, 6))  # Output: 2
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