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Algorithmica

, Volume 68, Issue 2, pp 448–482 | Cite as

Relative Convex Hulls in Semi-Dynamic Arrangements

  • Mashhood Ishaque
  • Csaba D. Tóth
Article

Abstract

We present a data structure for maintaining the geodesic hull of a set of points (sites) in the presence of pairwise noncrossing line segments (barriers) that subdivide a bounding box into simply connected faces. For m barriers and n sites, our data structure has O((m+n)logn) size. It supports a mixed sequence of O(m) barrier insertions and O(n) site deletions in \(O((m+n) \operatorname{polylog}(mn))\) total time, and answers analogues of standard convex hull queries in \(O(\operatorname{polylog}(mn))\) time.

Our data structure supports a generalization of the sweep line technique, in which the sweep wavefront is a simple closed polygonal curve, and it sweeps a set of n points in the plane by simple moves. We reduce the total time of supporting m online moves of a polygonal wavefront sweep algorithm from the naïve \(O(m\sqrt{n} \operatorname{polylog}n)\) to \(O((m+n) \operatorname{polylog}(mn))\).

Keywords

Relative convex hull Semi-dynamic data structure Plane sweep 

Notes

Acknowledgements

We are grateful to the anonymous referees for their many constructive suggestions that helped clarify the definitions and significantly improve the presentation.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceTufts UniversityMedfordUSA
  2. 2.Department of MathematicsUniversity of CalgaryCalgaryCanada

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