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Efficient Algorithms to Test Digital Convexity

  • Loïc CrombezEmail author
  • Guilherme D. da Fonseca
  • Yan Gérard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)

Abstract

A set \(S \subset \mathbb {Z}^d\) is digital convex if \({{\,\mathrm{conv}\,}}(S) \cap \mathbb {Z}^d = S\), where \({{\,\mathrm{conv}\,}}(S)\) 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 \(\varOmega (n \log n)\) time even for dimension \(d = 2\), we provide an algorithm for testing the digital convexity of \(S\subset \mathbb {Z}^2\) 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.

Keywords

Convexity Digital geometry 

Notes

Acknowledgement

This work has been sponsored by the French government research program “Investissements d’Avenir” through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Loïc Crombez
    • 1
    Email author
  • Guilherme D. da Fonseca
    • 1
  • Yan Gérard
    • 1
  1. 1.Université Clermont Auvergne and LIMOSClermont-FerrandFrance

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