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Algorithmica

, Volume 78, Issue 3, pp 990–1019 | Cite as

Connectivity Graphs of Uncertainty Regions

  • Erin Chambers
  • Alejandro Erickson
  • Sándor P. Fekete
  • Jonathan Lenchner
  • Jeff Sember
  • Venkatesh Srinivasan
  • Ulrike Stege
  • Svetlana Stolpner
  • Christophe Weibel
  • Sue Whitesides
Article

Abstract

We study connectivity relations among points, where the precise location of each input point lies in a region of uncertainty. We distinguish two fundamental scenarios under which uncertainty arises. In the favorable Best-Case Uncertainty, each input point can be chosen from a given set to yield the best possible objective value. In the unfavorable Worst-Case Uncertainty, the input set has worst possible objective value among all possible point locations, which are uncertain due, for example, to imprecise data. We consider these notions of uncertainty for the bottleneck spanning tree problem, giving rise to the following Best-Case Connectivity with Uncertainty problem: given a family of geometric regions, choose one point per region, such that the longest edge length of an associated geometric spanning tree is minimized. We show that this problem is NP-hard even for very simple scenarios in which the regions are line segments or squares. On the other hand, we give an exact solution for the case in which there are \(n+k\) regions, where k of the regions are line segments and n of the regions are fixed points. We then give approximation algorithms for cases where the regions are either all line segments or all unit discs. We also provide approximation methods for the corresponding Worst-Case Connectivity with Uncertainty problem: Given a set of uncertainty regions, find the minimal distance r such that for any choice of points, one per region, there is a spanning tree among the points with edge length at most r.

Keywords

Uncertainty Geometric optimization Connectivity Worst-case connectivity Best-case connectivity 

Notes

Acknowledgments

We are grateful for two Bellairs workshops supporting this research: the 8th and 9th McGill-INRIA-UVictoria Workshop on Computational Geometry in 2009 and 2010. We also thank the anonymous reviewers for many helpful and constructive comments that greatly helped to improve the overall presentation. We also acknowledge financial support by a number of different agencies, as follows. Erin Chambers was supported by NSF Grants CCF 1054779 and IIS 1319573. Alejandro Erickson was supported by the EPSRC, Grant No. EP/K015680/1. Ulrike Stege was supported by an NSERC Discovery Grant. Svetlana Stolpner was supported by the Fonds québécois de la recherche sur la nature et les technologies (FQRNT). Venkatesh Srinivasan was supported by an NSERC Discovery Grant. Sue Whitesides was supported by an NSERC Discovery Grant.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Erin Chambers
    • 1
  • Alejandro Erickson
    • 2
  • Sándor P. Fekete
    • 3
  • Jonathan Lenchner
    • 4
  • Jeff Sember
    • 5
  • Venkatesh Srinivasan
    • 6
  • Ulrike Stege
    • 6
  • Svetlana Stolpner
    • 7
  • Christophe Weibel
    • 8
  • Sue Whitesides
    • 6
  1. 1.Department of Computer ScienceSt. Louis UniversitySt. LouisUSA
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  3. 3.Department of Computer ScienceBraunschweig University of TechnologyBraunschweigGermany
  4. 4.IBM Research AfricaNairobiKenya
  5. 5.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  6. 6.Department of Computer ScienceUniversity of VictoriaVictoriaCanada
  7. 7.A.U.G. Signals Ltd.TorontoCanada
  8. 8.GoogleZurichSwitzerland

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