A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-Off Algorithms

Abstract

Given a read-only memory for input and a write-only stream for output, an s-workspace algorithm, for a positive integer parameter s, is an algorithm using only O(s) words of workspace in addition to the memory for the input. In this paper, we present an \(O(n^2/s)\)-time s-workspace algorithm for subdividing a simple n-gon into \(O(\min \{n/s,s\})\) subpolygons of complexity \(O(\max \{n/s,s\})\). As applications of the subdivision, the previously best known time-space trade-offs for the following three geometric problems are improved immediately by adopting the proposed subdivision: (1) computing the shortest path between two points inside a simple n-gon, (2) computing the shortest-path tree from a point inside a simple n-gon, (3) computing a triangulation of a simple n-gon. In addition, we improve the algorithm for problem (2) further by applying different approaches depending on the size of the workspace.

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Notes

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    We say we encounter an extension during the traversal of \(\partial P\) if we reach a foot point or the defining vertex of the extension.

References

  1. 1.

    Ahn, H.-K., Baraldo, N., Oh, E., Silvestri, F.: A time-space trade-off for triangulations of points in the plane. In: Proceedings of the 23rd Annual International Computing and Combinatorics Conference (COCOON 2017), pp. 3–12 (2017)

  2. 2.

    Aronov, B., Korman, M., Pratt, S., van Renssen, A., Roeloffzen, M.: Time-space trade-offs for triangulating a simple polygon. J. Comput. Geom. 8(1), 105–124 (2017)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Asano, T., Buchin, K., Buchin, M., Korman, M., Mulzer, W., Rote, G., Schulz, A.: Memory-constrained algorithms for simple polygons. Comput. Geom. 46(8), 959–969 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Asano, T., Kirkpatrick, D.: Time-space tradeoffs for all-nearest-larger-neighbors problems. In: Proceedings of the 13th Algorithms and Data Structures Symposium (WADS 2013), pp. 61–72 (2013)

  5. 5.

    Asano, T., Mulzer, W., Rote, G., Wang, Y.: Constant-work-space algorithms for geometric problems. J. Comput. Geom. 2(1), 46–68 (2011)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Asano, T., Mulzer, W., Wang, Y.: Constant-work-space algorithms for shortest paths in trees and simple polygons. J. Graph Algorithms Appl. 15(5), 569–586 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Banyassady, B., Barba, L., Mulzer, W.: Time-space trade-offs for computing Euclidean minimum spanning trees. In: Proceedings of the 13th Latin American Theoretical Informatics Symposium (LATIN 2018), pp. 108–119 (2018)

  8. 8.

    Banyassady, B., Korman, M., Mulzer, W.: Computational geometry column 67. SIGACT News 49(2), 77–94 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Barba, L., Korman, M., Langerman, S., Sadakane, K., Silveira, R.I.: Space-time trade-offs for stack-based algorithms. Algorithmica 72(4), 1097–1129 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(3), 485–524 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Darwish, O., Elmasry, A.: Optimal time-space tradeoff for the 2D convex-hull problem. In: Proceedings of the 22nd Annual European Symposium on Algorithms (ESA 2014), pp. 284–295 (2014)

  12. 12.

    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer TELOS, Berlin (2008)

    Book  MATH  Google Scholar 

  13. 13.

    Frederickson, G.N.: Upper bounds for time-space trade-offs in sorting and selection. J. Comput. Syst. 34(1), 19–26 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Guibas, L., Hershberger, J.: Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci. 39(2), 126–152 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2(1), 209–233 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Har-Peled, S.: Shortest path in a polygon using sublinear space. J. Comput. Geom. 7(2), 19–45 (2015)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Korman, M., Mulzer, W., van Renssen, A., Roeloffzen, M., Seiferth, P., Stein, Y.: Time-space trade-offs for triangulations and Voronoi diagrams. Comput. Geom. 73, 35–45 (2018)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Hee-Kap Ahn.

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This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the SW Starlab support program (IITP-2017-0-00905) supervised by the IITP (Institute for Information & Communications Technology Promotion).

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Oh, E., Ahn, HK. A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-Off Algorithms. Algorithmica 81, 2829–2856 (2019). https://doi.org/10.1007/s00453-019-00558-9

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Keywords

  • Time-space trade-off
  • Balanced subdivision
  • Simple polygon
  • Shortest path
  • Shortest path tree
  • Triangulation