Dynamic ham-sandwich cuts in the plane

Abstract

We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham-sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O ( n 1 / 3 + ε ) amortized time and O ( n 4 / 3 + ε ) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O ( log n ) time and ham-sandwich queries in O ( k log 4 n ) time. In addition, if each point set has convex peeling depth k, then we can maintain the decomposition automatically using O ( k log n ) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O ( k log 4 n ) time plus the O ( ( k b ) polylog ( k b ) ) time required to approximate the root of a polynomial of degree O ( k ) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.