A First Algorithm: Planar Convex Hulls

Adaptive Planar Convex Hull Algorithm for a Set of Convex Hulls

Optimal, Output-sensitive Algorithms for Constructing Planar Hulls in Parallel

In this paper we focus on the problem of designing very fast parallel algorithms for the planar convex hull problem that achieve the optimal O(n log H) work-bound for input size n and output size H. Our algorithms are designed for the arbitrary CRCW PRAM model. We first describe a very simple ©(log n log H) time optimal deterministic algorithm for the planar hulls which is an improvement over t...

Self-Improving Algorithms for Coordinatewise Maxima and Convex Hulls

Finding the coordinate-wise maxima and the convex hull of a planar point set are probably the most classic problems in computational geometry. We consider these problems in the selfimproving setting. Here, we have n distributions D1, . . . ,Dn of planar points. An input point set (p1, . . . , pn) is generated by taking an independent sample pi from each Di, so the input is distributed according...

Minimum convex partition of a constrained point set

A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition with respect to S is a convex partition of S such that the number of convex polygons is minimised. In this paper, we wil...

Convex Hulls of Multidimensional Random Walks

Let Sk be a random walk in R such that its distribution of increments does not assign mass to hyperplanes. We study the probability pn that the convex hull conv(S1, . . . , Sn) of the first n steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, pn does not depend on the distribution of increments. This e...