Streaming Points: Remove Any Pair Within a Distance

Problem Statement

You are building a system that processes a stream of 1D points on a number line. You need to implement a class PointProcessor that supports two operations: adding a new point and removing a point. When a new point is added, if there is an existing point P in the system such that the distance |new_point - P| <= D (where D is a predefined threshold), you must remove P from the system instead of adding the new point. If multiple such points exist, you can remove any one of them. If no such point exists, add the new point to the system.

Interface

class PointProcessor:
    def __init__(self, d: float):
        """Initializes the processor with distance threshold d."""
        self.d = d

    def process(self, point: float) -> bool:
        """ 
        If a point exists in the system within distance d, 
        remove it and return False. Otherwise, add the point 
        and return True.
        """

Examples

Example 1:

• D = 2.0
• process(5.0) -> returns True (System: {5.0})
• process(6.0) -> returns False (Distance |6.0 - 5.0| = 1.0 True(System:{10.0}`)
• process(12.0) -> True (System: {10.0, 12.0})
• process(11.0) -> False (Distance to 10.0 is 1.0, remove 10.0, System: {12.0})

Constraints

• Number of operations: up to 10^5.
• Points are floating-point numbers.
• Time complexity should be better than O(N) per operation.

Edge Cases

• Points with the exact same value.
• Multiple points within range D (the problem allows removing any valid candidate).