Given a set of intervals, find the interval which has the maximum number of intersections

Question

Given a set of intervals, find the interval which has the maximum number of intersections (not the length of a particular intersection). So if input (1,6) (2,3) (4,11), (1,6

Answer 1

Note: David Eisenstat's algorithm has better performance than this one.

A simple plane-sweep algorithm will do this in O(nlogn + m*n), where m is the maximum number of intersections with any single interval.

Sort the interval endpoints. Keep track of the active segments. When reaching a start point, increment the intersection counts of all active intervals, and set the new interval's intersection count to the number of active intervals (excluding itself). When reaching an end point, remove the interval from the active intervals.

Answer 2

Don't use an interval tree. Create an event for each interval endpoint, so that each interval has a start event and a stop event. Process these events in order. (Order starts before stops if a measure-zero intersection counts as an intersection, or stops before starts otherwise.)

Initialize a map C from intervals to numbers. Initialize the start count s = 0 and the stop count t = 0. To process a start event for interval i, set s = s + 1 and then C(i) = -(t + 1). To process a stop event for interval i, set t = t + 1 and then C(i) = C(i) + s.

At the end, C maps intervals to their intersection counts. This algorithm is O(n log n) because of the sorting; that running time is optimal if the endpoints can be added and compared only, by fairly standard computational geometry techniques.