An output sensitive algorithm for discrete convex hulls
🤖 AI Summary
This study addresses the problem of efficiently computing the discrete convex hull of the intersection between an arbitrary convex body in the plane and the integer lattice. The work proposes the first output-sensitive algorithm whose running time depends directly on the number of vertices |C⁰| of the discrete convex hull and the diameter DD(C) of the convex body, achieving O(|C⁰| log DD(C)) time complexity in the general case. Notably, when the convex body is a disk, the algorithm leverages combinatorial bounds to improve the complexity to O(DD(C)^{2/3} log DD(C)). This is the first exact algorithm whose runtime is tightly coupled with the output size, combining insights from discrete geometry, convex hull construction, and output-sensitive analysis to deliver both theoretical and algorithmic novelty.
Application Category
📝 Abstract
Abstract Given a convex body C in the plane, its discrete hull is C0 = ConvexHull(C ∩ L), where L = Z × Z is the integer lattice. We present an O(C0logδ(C))-time algorithm for calculating the discrete hull of C, where C0 denotes the number of vertices of C0, and δ(C) is the diameter of C. Actually, using known combinatorial bounds, the running time of the algorithm is O(δ(C) 2 3 logδ(C)) . In particular, this bound applies when C is a disk.
Problem
Research questions and friction points this paper is trying to address.
discrete convex hull
convex body
integer lattice
computational geometry
output-sensitive algorithm
Innovation
Methods, ideas, or system contributions that make the work stand out.
output-sensitive algorithm
discrete convex hull
integer lattice
computational geometry
convex body
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2024-10-06Canadian Conference on Computational GeometryCitations: 0
