An output sensitive algorithm for discrete convex hulls

1998-05-01

SCG '98

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.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationPlanning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the webSecurity and Privacy: Large-scale security measurements

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(C0logδ(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

• discrete convex hull
• convex body
• integer lattice
• computational geometry
• output-sensitive algorithm

Innovation

• output-sensitive algorithm
• discrete convex hull
• integer lattice
• computational geometry
• convex body