Took me a minute to strip away the space theming and realize this was just a merge intervals problem.
Model each gravitational zone as an interval [center - radius, center + radius] on the position line. Merge all overlapping or adjacent intervals to form the union of unsafe regions, then compute the complement intervals as the safe ranges. Handle edge cases like empty input or zones covering the entire line.
Pro tip: Clarify whether the boundaries are inclusive or exclusive and whether positions are discrete or continuous; this affects whether touching intervals merge and how you output the safe ranges. Also, mention that sorting by start point enables an O(n log n) solution, which is optimal for large inputs.
Ask about input format, whether positions are integers or real numbers, boundary inclusivity, and what to return if no safe ranges exist. Confirm if the list can be empty or contain invalid pairs.
For each pair (center, radius), compute the unsafe interval [center - radius, center + radius]. Store these intervals in a list.
Sort intervals by start point. Iterate through them, merging overlapping or adjacent intervals into a list of disjoint unsafe intervals.
Given the merged unsafe intervals, derive the safe intervals by taking the gaps between them, including from -infinity to the first start and from the last end to +infinity if applicable.
Return the list of safe ranges. Analyze time complexity O(n log n) due to sorting and space complexity O(n) for storing intervals.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.