Dynamic Ham-Sandwich Cuts for Two Overlapping Point Sets

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two overlapping point sets in convex position in the plane. The ham-sandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports insertion and deletion of vertices in O(log n) time, and area, perimeter and vertex-count queries in O(log n) time, where n is the total number of points of S1 ∪ S2. An extension of the algorithm for sets with a bounded number c of convex hull peeling layers enables area and perimeter queries, using O(c log n) time for insertions and deletions and O(log n) query time. Our algorithm improves previous results [15, 1]. The static method of Stojmenović [15] and the dynamic algorithm of Abbott et al. [1] maintain a ham-sandwich cut of two disjoint convex polygons in the plane. Our dynamic algorithm removes the restrictions based on the separation of the points. It also solves an open problem from 1991 about finding the area and perimeter ham-sandwich cuts for overlapping convex point sets in the static setting [15].