algoadvance

You are given a 2D integer array ranges and two integers left and right. Each ranges[i] = [start_i, end_i] represents an inclusive interval.

Return true if each integer in the inclusive interval [left, right] is covered by at least one interval in ranges. Return false otherwise.

Clarifying Questions

  1. Will the range intervals in ranges be sorted or unordered?
    • The intervals can be unordered.
  2. Are the intervals in ranges distinct or can they overlap?
    • The intervals can overlap.
  3. What is the size limit for ranges and the range for left and right?
    • We assume they fit within the constraints typical for a competitive programming problem (e.g., a few thousand intervals).

Strategy

Our goal is to check if every integer from left to right is included in at least one of the intervals in ranges. Here is a step-by-step strategy:

  1. Create a boolean list covered initialized to False with a size covering from left to right.
  2. Iterate through each interval in ranges and mark the corresponding elements in covered as True.
  3. Check if all elements in the covered list corresponding to the range [left, right] are True.

We can initialize a boolean array representing the coverage status of each number in the range [left, right]. For each range in ranges, mark the appropriate indices as covered. Finally, validate that all indices corresponding to [left, right] are True.

Code

def isCovered(ranges, left, right):
    # Create a list to keep track of which numbers from left to right are covered.
    covered = [False] * (right - left + 1)
    
    # Mark the numbers covered by the ranges
    for start, end in ranges:
        for num in range(max(start, left), min(end, right) + 1):
            covered[num - left] = True
    
    # Check if all numbers from left to right are covered
    return all(covered)

# Example usage:
ranges = [[1,2], [3,4], [5,6]]
left = 2
right = 5
print(isCovered(ranges, left, right))  # Output: True

Strategy Explanation

  1. Creating covered List:
    • The length of the covered list is the size of the interval [left, right], hence it has (right - left + 1) elements.
  2. Marking Coverage:
    • For each interval [start, end], we iterate only over the intersection with [left, right].
    • max(start, left) ensures we don’t start before left.
    • min(end, right) ensures we don’t go past right.
    • Mark covered[num - left] as True where num is within the intersection.
  3. Checking Coverage:
    • The all(covered) function checks if all elements in the covered list are True.

Time Complexity

This solution should be efficient enough given typical problem constraints.

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