An output sensitive algorithm for discrete convex hulls
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.