Dynamic Ham-Sandwich Cuts for Two Point Sets with Bounded Convex-Hull-Peeling Depth

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two (possibly overlapping) point sets in the plane, with a bounded number of convex-hull peeling layers. 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(c log n) time, area and perimeter queries in O(log n) time and vertex-count queries in O(c log n) time, where n is the total number of points of S1 ∪ S2 and c is a bound on the number of convex hull peeling layers. Our algorithm considerably improves previous results [15, 1]. Stojmenović’s [15] static method finds a ham-sandwich cut for two separated point sets. The dynamic algorithm of Abbott et al. [1] maintains a ham-sandwich cut of two disjoint convex polygons in the plane. Our dynamic algorithm removes the restrictions based on the convex position and 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].