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Describing Multivariate Distributions with Nonlinear Variation Using Data Depth
, 2006
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Dynamic HamSandwich Cuts for Two Point Sets with Bounded ConvexHullPeeling Depth
"... We provide an efficient data structure for dynamically maintaining a hamsandwich cut of two (possibly overlapping) point sets in the plane, with a bounded number of convexhull peeling layers. The hamsandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex coun ..."
Abstract

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We provide an efficient data structure for dynamically maintaining a hamsandwich cut of two (possibly overlapping) point sets in the plane, with a bounded number of convexhull peeling layers. The hamsandwich 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 vertexcount queries in O(c 3 log 3 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 hamsandwich cut for two separated point sets. The dynamic algorithm of Abbott et al. [1] maintains a hamsandwich 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 hamsandwich cuts for overlapping convex point sets in the static setting [15]. 1
Dynamic HamSandwich Cuts for Two Overlapping Point Sets
"... We provide an efficient data structure for dynamically maintaining a hamsandwich cut of two overlapping point sets in convex position in the plane. The hamsandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports ..."
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We provide an efficient data structure for dynamically maintaining a hamsandwich cut of two overlapping point sets in convex position in the plane. The hamsandwich 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 vertexcount queries in O(log 3 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 3 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 hamsandwich 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 hamsandwich cuts for overlapping convex point sets in the static setting [15]. 1