Efficient Algorithms to Test Digital Convexity

Abstract : A set S ⊂ Z^d is digital convex if conv(S) ∩ Z^d = S, where conv(S) denotes the convex hull of S. In this paper, we consider the algorithmic problem of testing whether a given set S of n lattice points is digital convex. Although convex hull computation requires Ω(n log n) time even for dimension d = 2, we provide an algorithm for testing the digital convexity of S ⊂ Z^2 in O(n + h log r) time, where h is the number of edges of the convex hull and r is the diameter of S. This main result is obtained by proving that if S is digital convex, then the well-known quickhull algorithm computes the convex hull of S in linear time. In fixed dimension d, we present the first polynomial algorithm to test digital convexity, as well as a simpler and more practical algorithm whose running time may not be polynomial in n for certain inputs.
Document type :
Conference papers
Complete list of metadatas

Cited literature [27 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01983460
Contributor : Loïc Crombez <>
Submitted on : Wednesday, January 16, 2019 - 2:28:46 PM
Last modification on : Monday, January 20, 2020 - 12:14:05 PM

File

main.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01983460, version 1

Collections

Citation

Loïc Crombez, Guilherme da Fonseca, Yan Gerard. Efficient Algorithms to Test Digital Convexity. 21st IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI 2019, Mar 2019, Paris, France. ⟨hal-01983460⟩

Share

Metrics

Record views

59

Files downloads

128