, 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


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.


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



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.


  1. 1.
    Abellanas, M., Hurtado, F., Ramos, P.: Structural tolerance and Delaunay triangulation. Inf. Process. Lett. 71, 221–227 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alt, H., Arkin, E., Brönnimann, H., Erickson, J., Fekete, S., Knauer, C., Lenchner, J., Mitchell, J., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Proceedings of the 22nd ACM Symposium on Computational Geometry (SoCG), pp. 449–458 (2006)Google Scholar
  3. 3.
    Arkin, E.M., Dieckmann, C., Knauer, C., Mitchell, J.S.B., Polishchuk, V., Schlipf, L., Yang, S.: Convex transversals. In: Proceedings of the 12th International Symposium on Algorithms and Data Structures (WADS), pp. 49–60 (2011)Google Scholar
  4. 4.
    Arkin, E.M., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Appl. Math. 55(3), 197–218 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chambers, E.W., Erickson, A., Fekete, S.P., Lenchner, J., Sember, J., Srinivasan, V., Stege, U., Stolpner, S., Weibel, C., Whitesides, S.: Connectivity graphs of uncertainty regions. In: 21st International Symposium on Algorithms and Computation (ISAAC), number 5307 in Springer LNCS, pp. 434–445 (2010)Google Scholar
  6. 6.
    Clementi, A.E.F., Penna, P., Silvestri, R.: On the power assignment problem in radio networks. Technical Report TR00-054, Electronic Colloquium on Computational Complexity (2000)Google Scholar
  7. 7.
    de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. J. Algorithms 57(1), 22–36 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ding, H., Xu, J.: Solving the chromatic cone clustering problem via minimum spanning sphere. In: 38th International Colloquium on Automata, Languages and Programming (ICALP), volume 6755 of Springer LNCS, pp. 773–784 (2011)Google Scholar
  9. 9.
    Disser, Y., Mihalák, M., Montanari, S.: Max shortest path for imprecise points. In: 31st European Workshop on Computational Geometry, pp. 184–187 (2015)Google Scholar
  10. 10.
    Disser, Y., Mihalák, M., Montanari, S., Widmayer, P.: Rectilinear shortest path and rectilinear minimum spanning tree with neighborhoods. In: Proceedings of the 3rd International Symposium on Combinatorial Optimization (ISCO), pp. 208–220 (2014)Google Scholar
  11. 11.
    Dorrigiv, R., Fraser, R., He, M., Kamali, S., Kawamura, A., López-Ortiz, A., Seco, D.: On minimum- and maximum-weight minimum spanning trees with neighborhoods. In: Proceedings of the 10th Workshop on Approximation and Online Algorithms (WAOA), pp. 93–106 (2012)Google Scholar
  12. 12.
    Dror, M., Efrat, A., Lubiw, A., Mitchell, J.S.B.: Touring a sequence of polygons. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pp. 473–482 (2003)Google Scholar
  13. 13.
    Duchet, P., Hamidoune, Y.O., Vergnas, M.L., Meyniel, H.: Representing a planar graph by vertical lines joining different levels. Discrete Math. 46(3), 319–321 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. In: Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 38–46 (2001)Google Scholar
  15. 15.
    Fiala, J., Kratochvíl, J., Proskurowski, A.: Systems of distant representatives. Discrete Appl. Math. 145(2), 306–316 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fuchs, B.: On the hardness of range assignment problems. In: Proceedings of the 6th Italian Conference on Algorithms and Complecity (CIAC), pp. 127–138 (2006)Google Scholar
  17. 17.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  18. 18.
    Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods. Nord. J. Comput. 6(4), 469–488 (1999)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lev-Tov, N., Peleg, D.: Exact algorithms and approximation schemes for base station placement problems. In: Proceedings of the 8th Scandinavian Workshop Algorithm Theory (SWAT), pp. 90–99 (2002)Google Scholar
  20. 20.
    Lev-Tov, N., Peleg, D.: Polynomial time approximation schemes for base station coverage with minimum total radii. Comput. Netw. 47(4), 489–501 (2005)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Löffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica 56, 235–269 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Löffler, M., van Kreveld, M.: Largest bounding box, smallest diameter, and related problems on imprecise points. Comput. Geom. Theory Appl. 43, 419–433 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mata, C., Mitchell, J.: Approximation algorithms for geometric tour and network design problems. In: Proceedings of the 11th ACM Symposium on Computational Geometry (SoCG), pp. 360–369 (1995)Google Scholar
  25. 25.
    Mitchell, J.S.B.: A PTAS for TSP with neighborhoods among fat regions in the plane. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 11–18 (2007)Google Scholar
  26. 26.
    Pan, X., Li, F., Klette, R.: Approximate shortest path algorithms for sequences of pairwise disjoint simple polygons. In: Proceedings of the 22nd Canadian Conference on Computational Geometry (CCCG), pp. 175–178 (2010)Google Scholar
  27. 27.
    Parker, G., Rardin, R.L.: Guaranteed performance heuristics for the bottleneck traveling salesman problem. Oper. Res. Lett. 2(6), 269–272 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yang, Y., Lin, M., Xu, J., Xie, Y.: Minimum spanning tree with neighborhoods. In: Proceedings of the 3rd Conference on Algorithmic Aspects on Information and Management (AAIM), pp. 306–316. Springer, Berlin, Heidelberg (2007)Google Scholar

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

Personalised recommendations