Skip to main content
Log in

Spanning simple path inside a simple polygon

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

Given a set S of n colored points of m colors inside a simple polygon P, each point within the polygon has a specific color that is not necessarily unique, i.e., they may exhibit the same color. The study aims to find a simple path that traverses at least one point of each color using a set of S points contained within a simple polygon P. Two results are presented in this study. First, we demonstrate that finding such simple paths inside a simple polygon is an \(NP-complete\) problem. Moreover, we provide a polynomial-time algorithm that computes the simple path when P is an orthogonal spiral simple polygon, and our objective is to locate a simple Hamiltonian path L using all points of S inside P. Our algorithm has a time complexity of \(O(r+rn^4)\), where r is the number of reflex vertices in P and n is the number of points in S.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Cheng Q, Chrobak M, Sundaram G (2000) Computing simple paths among obstacles. Comput Geom 16(4):223–233. https://doi.org/10.1016/S0925-7721(00)00011-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Daescu O, Luo J (2008) Computing simple paths on points in simple polygons. In: Ito H, Kano M, Katoh N, Uno Y (eds) Comput Geom Graph Theory. Springer, Berlin, Heidelberg, pp 41–55

    Chapter  Google Scholar 

  3. Tan X, Jiang B (2019) Computing simple paths from given points inside a polygon. Discret Appl Math 252:67–76. https://doi.org/10.1016/j.dam.2018.09.020

    Article  MathSciNet  MATH  Google Scholar 

  4. Fadavian M, Fadavian H (2022) A genetic algorithm for straight-line embedding of a cycle onto a given set of points inside the general simple polygons. arXiv preprint arXiv:2203.00453

  5. Razzazi M, Sepahvand A (2017) Time complexity of two disjoint simple paths. Scientia Iranica 24(3):1335–1343. https://doi.org/10.24200/sci.2017.4116

    Article  Google Scholar 

  6. Schüler J, Spillner A (2014) Crossing-free spanning trees in visibility graphs of points between monotone polygonal obstacles. In: Hirsch EA, Kuznetsov SO, Pin J-É, Vereshchagin NK (eds) Comput Sci - Theory Appl. Springer, Cham, pp 337–350

    Google Scholar 

  7. Alsuwaiyel MH, Lee DT (1993) Minimal link visibility paths inside a simple polygon. Comput Geom 3(1):1–25. https://doi.org/10.1016/0925-7721(93)90027-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Erickson LH, LaValle SM (2013) A simple, but np-hard, motion planning problem. In: Twenty-Seventh AAAI Conference on Artificial Intelligence

  9. Keshavarz-Kohjerdi F, Bagheri A (2017) A linear-time algorithm for finding hamiltonian (s, t)-paths in odd-sized rectangular grid graphs with a rectangular hole. J Supercomput 73(9):3821–3860

    Article  MATH  Google Scholar 

  10. Keshavarz-Kohjerdi F, Bagheri A (2013) An efficient parallel algorithm for the longest path problem in meshes. —————————Alert—————————Updated—————————OK——————————————————Alert—————————Updated—————————OK——————————————————Alert—————————Updated—————————OK—————————J Supercomput 65(2):723–741

    Google Scholar 

  11. Consuegra ME, Narasimhan G (2013) Geometric Avatar Problems. In: Seth A, Vishnoi NK (eds) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013), vol 24. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, pp 389–400

    Google Scholar 

  12. Manzini R, Gamberini R (2008) Design, management and control of logistic distribution systems. Int J Adv Robot Syst 1:263–290

    Google Scholar 

  13. Acharyya A, Nandy SC, Roy S (2018) Minimum width color spanning annulus. Theor Comput Sci 725:16–30

    Article  MathSciNet  MATH  Google Scholar 

  14. Abellanas M, Hurtado F, Icking C, Klein R, Langetepe E, Ma L, Palop B, Sacristán V (2001) Smallest color-spanning objects. In: European Symposium on Algorithms, pp. 278–289. Springer

  15. Acharyya A, Jallu RK, Keikha V, Löffler M, Saumell M (2021) Minimum color spanning circle in imprecise setup. In: International Computing and Combinatorics Conference, pp. 257–268. Springer

  16. Das S, Goswami PP, Nandy SC (2009) Smallest color-spanning object revisited. Int J Comput Geom Appl 19(05):457–478

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang M, Wang H (2014) Shortest color-spanning intervals. In: International Computing and Combinatorics Conference, pp. 288–299. Springer

  18. Hasheminejad J, Khanteimouri P, Mohades A (2015) Computing the smallest color spanning equilateral triangle. Proceedings of 31st EuroCG, pp 32–35

  19. Khanteimouri P, Mohades A, Abam MA, Kazemi MR (2013) Computing the smallest color-spanning axis-parallel square. In: International Symposium on Algorithms and Computation, pp. 634–643. Springer

  20. Keikha V, Keikha H, Mohades A (2021) On the k-colored rainbow sets in fixed dimensions. In: International Conference on Combinatorial Optimization and Applications, pp. 587–601. Springer

  21. Pruente J (2019) Minimum diameter color-spanning sets revisited. Discrete Optim 34:100550

    Article  MathSciNet  MATH  Google Scholar 

  22. Fleischer R, Xu X (2011) Computing minimum diameter color-spanning sets is hard. Inform Proc Lett 111(21–22):1054–1056

    Article  MathSciNet  MATH  Google Scholar 

  23. Bilge YC, Çağatay D, Genç B, Sarı M, Akcan H, Evrendilek C (2015) All colors shortest path problem. arXiv preprint arXiv:1507.06865

  24. Akçay MB, Akcan H, Evrendilek C (2018) All colors shortest path problem on trees. J Heuristics 24(4):617–644

    Article  Google Scholar 

  25. Carrabs F, Cerulli R, Pentangelo R, Raiconi A (2018) A two-level metaheuristic for the all colors shortest path problem. Comput Optim Appl 71(2):525–551

    Article  MathSciNet  MATH  Google Scholar 

  26. Carrabs F, Cerulli R, Raiconi A (2021) A reduction heuristic for the all-colors shortest path problem. RAIRO-Op Res 55:2071–2082

    Article  MathSciNet  MATH  Google Scholar 

  27. Yannakakis M (1978) Node-and edge-deletion np-complete problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pp. 253–264

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mohammadreza Razzazi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sepahvand, A., Razzazi, M. Spanning simple path inside a simple polygon. J Supercomput 79, 2740–2766 (2023). https://doi.org/10.1007/s11227-022-04765-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-022-04765-0

Keywords

Navigation