research-article

Computing hereditary convex structures

Authors:

Bernard Chazelle,

Wolfgang MulzerAuthors Info & Claims

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

Pages 61 - 70

https://doi.org/10.1145/1542362.1542374

Published: 08 June 2009 Publication History

Abstract

Color red and blue the n vertices of a convex polytope P in R³. Can we compute the convex hull of each color class in o(n log n)? What if we have k > 2 colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eight-year old open problem by showing how to split a convex polytope in linear expected time.

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Cited By

Buchin KLöffler MMorin PMulzer W(2018)Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and ExtendedAlgorithmica10.1007/s00453-010-9430-061:3(674-693)Online publication date: 31-Dec-2018
https://dl.acm.org/doi/10.1007/s00453-010-9430-0
Löffler MMulzer WRandall D(2011)Triangulating the square and squaring the triangleProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133171(1759-1777)Online publication date: 23-Jan-2011
https://dl.acm.org/doi/10.5555/2133036.2133171
Ezra EMulzer WHurtado Fvan Kreveld M(2011)Convex hull of imprecise points in o(n log n) time after preprocessingProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998199(11-20)Online publication date: 13-Jun-2011
https://dl.acm.org/doi/10.1145/1998196.1998199
Show More Cited By

Index Terms

Computing hereditary convex structures
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

June 2009

426 pages

ISBN:9781605585017

DOI:10.1145/1542362

Program Chairs:
John Hershberger
Mentor Graphics Corp.
,
Efi Fogel
Tel Aviv University

Copyright © 2009 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 08 June 2009

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SoCG '09: 25th Annual Symposium on Computational Geometry

June 8 - 10, 2009

Aarhus, Denmark

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Buchin KLöffler MMorin PMulzer W(2018)Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and ExtendedAlgorithmica10.1007/s00453-010-9430-061:3(674-693)Online publication date: 31-Dec-2018
https://dl.acm.org/doi/10.1007/s00453-010-9430-0
Löffler MMulzer WRandall D(2011)Triangulating the square and squaring the triangleProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133171(1759-1777)Online publication date: 23-Jan-2011
https://dl.acm.org/doi/10.5555/2133036.2133171
Ezra EMulzer WHurtado Fvan Kreveld M(2011)Convex hull of imprecise points in o(n log n) time after preprocessingProceedings of the twenty-seventh annual symposium on Computational geometry10.1145/1998196.1998199(11-20)Online publication date: 13-Jun-2011
https://dl.acm.org/doi/10.1145/1998196.1998199
Buchin KMulzer W(2011)Delaunay triangulations in O(sort(n)) time and moreJournal of the ACM10.1145/1944345.194434758:2(1-27)Online publication date: 11-Apr-2011
https://dl.acm.org/doi/10.1145/1944345.1944347
Buchin KMulzer W(2009)Delaunay Triangulations in O(sort(n)) Time and MoreProceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science10.1109/FOCS.2009.53(139-148)Online publication date: 25-Oct-2009
https://dl.acm.org/doi/10.1109/FOCS.2009.53
Buchin KLöffler MMorin PMulzer W(2009)Delaunay Triangulation of Imprecise Points Simplified and ExtendedProceedings of the 11th International Symposium on Algorithms and Data Structures10.1007/978-3-642-03367-4_12(131-143)Online publication date: 24-Jul-2009
https://dl.acm.org/doi/10.1007/978-3-642-03367-4_12

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