Dynamic Ham-Sandwich Cuts of Convex Polygons in the Plane
نویسندگان
چکیده
We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two nonoverlapping convex polygons in the plane. Given two non-overlapping convex polygons P1 and P2 in the plane, the ham-sandwich cut of P1 and P2 is a line that simultaneously bisects the area (or perimeter or vertex count) of both polygons. We provide a data structure that supports queries for the ham-sandwich cut in O(log n) worst-case time and insertions and deletions of vertices of the Pi in O(log n) worst-case time. We also show how this data structure can be used to maintain a partition of the plane by two lines into four regions each containing a quarter of the total polygon area (or perimeter or vertex count). In particular, if we use the vertex-count measure, the intersection of these two lines gives a point of Tukey depth n/4, which serves as an approximate Tukey median.
منابع مشابه
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...
متن کاملDynamic 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) tim...
متن کاملDynamic ham-sandwich cuts in the plane
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 hamsandwich 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) amortized time and O(n) space. Our second data st...
متن کاملFinding 2 d ham sandwich cuts in linear time ∗
A ham sandwich cut in d dimensions is a (d − 1)-dimensional hyperplane that divides each of d objects in half. While the existence of such a hyperplane was shown in 1938, little is known about how to find one. We are the first to show how this can be done in 2 dimensions when both objects are (possibly overlapping) convex polygons. Our algorithm runs in O(N) time where N is the sum of the numbe...
متن کامل