Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods

  • Yann Disser
  • Matúš Mihalák
  • Sandro Montanari
  • Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)


We consider a setting where we are given a graph \(\mathcal {G}=(\mathcal {R},E)\), where \(\mathcal {R}=\{R_1,\ldots ,R_n\}\) is a set of polygonal regions in the plane. Placing a point \(p_i\) inside each region \(R_i\) turns \(G\) into an edge-weighted graph \(G_{\varvec{p}}\), \({\varvec{p}}=\{p_1,\ldots ,p_n\}\), where the cost of \((R_i,R_j)\in E\) is the distance between \(p_i\) and \(p_j\). The Shortest Path Problem with Neighborhoods asks, for given \(R_s\) and \(R_t\), to find a placement \(\varvec{p}\) such that the cost of a resulting shortest \(st\)-path in \(\mathcal {G}_{\varvec{p}}\) is minimum among all graphs \(\mathcal {G}_{\varvec{p}}\). The Minimum Spanning Tree Problem with Neighborhoods asks to find a placement \(\varvec{p}\) such that the cost of a resulting minimum spanning tree is minimum among all graphs \(\mathcal {G}_{\varvec{p}}\). We study these problems in the \(L_1\) metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is \(\mathsf {APX}\)-hard, even if the neighborhood regions are segments.


Neighborhoods Minimum spanning tree Shortest path 



This work was supported by the EU FP7/2007-2013 (DG CONNECT.H5-Smart Cities and Sustainability), under grant agreement no. 288094 (project eCOMPASS) and by the Alexander von Humboldt-Foundation.


  1. 1.
    Ahadi, A., Mozafari, A., Zarei, A.: Touring disjoint polygons problem is NP-hard. In: Widmayer, P., Xu, Y., Zhu, B. (eds.) COCOA 2013. LNCS, vol. 8287, pp. 351–360. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  2. 2.
    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: Erlebach, T., Persiano, G. (eds.) WAOA 2012. LNCS, vol. 7846, pp. 93–106. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete algorithms (SODA), pp. 38–46 (2001)Google Scholar
  5. 5.
    Elbassioni, K.M., Fishkin, A.V., Mustafa, N.H., Sitters, R.A.: Approximation algorithms for euclidean group TSP. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1115–1126. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  6. 6.
    Knuth, D., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discrete Math. 5(3), 422–427 (1992)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Löffler, M., Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica 56(2), 235–269 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Löffler, M., van Kreveld, M.J.: Largest bounding box, smallest diameter, and related problems on imprecise points. Comput. Geom. 43(4), 419–433 (2010)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Yang, Y., Lin, M., Xu, J., Xie, Y.: minimum spanning tree with neighborhoods. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 306–316. Springer, Heidelberg (2007) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yann Disser
    • 1
  • Matúš Mihalák
    • 2
  • Sandro Montanari
    • 2
  • Peter Widmayer
    • 2
  1. 1.Department of MathematicsTU BerlinBerlinGermany
  2. 2.Department of Computer ScienceETH ZurichZurichSwitzerland

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