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Interval Intersection in O(1) Time Complexity using Python


Given two closed intervals [A1, B1] and [A2, B2], check and return if the two intervals intersect each other.

A closed interval [a, b] (with a <= b) denotes the set of real numbers x with a <= x <= b.

Two closed intervals are said to intersect if they have common numbers, i.e. that appear in both intervals.

Understanding the Problem

The core challenge of this problem is to determine if two intervals overlap. This is a common problem in computational geometry and has applications in scheduling, computer graphics, and more. A common pitfall is to overcomplicate the solution by considering all possible overlap scenarios, but a simpler approach exists.

Approach

To solve this problem, we can think about when the intervals do not intersect. Two intervals do not intersect if one is completely to the left of the other. This can be checked with the conditions:

  • B1 < A2: Interval 1 ends before Interval 2 starts.
  • B2 < A1: Interval 2 ends before Interval 1 starts.

If neither of these conditions is true, the intervals must intersect.

Naive Solution

A naive solution might involve checking all possible overlap scenarios, but this is unnecessary and inefficient. Instead, we can use the above conditions to simplify our solution.

Optimized Solution

The optimized solution involves a single if-statement to check the non-intersecting conditions. If either condition is true, the intervals do not intersect; otherwise, they do.

Algorithm

Here is a step-by-step breakdown of the algorithm:

  1. Check if B1 < A2 or B2 < A1.
  2. If either condition is true, return False.
  3. Otherwise, return True.

Code Implementation

def intervals_intersect(A1, B1, A2, B2):
    # Check if one interval is completely to the left of the other
    if B1 < A2 or B2 < A1:
        return False
    return True

Complexity Analysis

The time complexity of this solution is O(1) because it involves a constant number of operations. The space complexity is also O(1) as no additional space is used.

Edge Cases

Consider the following edge cases:

  • Intervals that touch at a single point, e.g., [1, 4] and [4, 6]. These should intersect.
  • Intervals that are completely separate, e.g., [1, 2] and [3, 4]. These should not intersect.

Testing

To test the solution comprehensively, consider a variety of test cases:

# Test cases
print(intervals_intersect(1, 6, 3, 8))  # True
print(intervals_intersect(1, 4, 4, 6))  # True
print(intervals_intersect(1, 8, 4, 6))  # True
print(intervals_intersect(4, 6, 1, 3))  # False
print(intervals_intersect(1, 2, 3, 4))  # False

Thinking and Problem-Solving Tips

When approaching such problems, consider the simplest conditions that define the problem. Avoid overcomplicating the solution. Practice by solving similar problems and studying different algorithms to improve problem-solving skills.

Conclusion

Understanding and solving interval intersection problems is crucial in many fields. The key is to simplify the problem by focusing on non-intersecting conditions. Practice and exploration of similar problems will enhance your problem-solving abilities.

Additional Resources