Skip to main content
SpringerLink
Log in

Solving the Problem of Decomposition of an Orthogonal Polyhedron of Arbitrary Dimension

  • Conference paper
  • First Online:
Advances in Mechanical Engineering (MMESE 2020)

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

Included in the following conference series:

Abstract

The paper is devoted to the problem of decomposition of an orthogonal polyhedron of arbitrary dimension into a set of large orthogonal objects. To solve this problem, an algorithm has been developed based on the application of the model of potential containers which is used in solving of orthogonal packing problems of arbitrary dimension. Examples of decomposition of two-dimensional and three-dimensional orthogonal polyhedrons are given.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Chekanin, V.A., Chekanin, A.V.: Algorithm for the placement of orthogonal polyhedrons for the cutting and packing problems. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 41–48. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39500-1_5

  2. Chekanin, V.A., Chekanin, A.V.: Development of algorithms for the correct visualization of two-dimensional and three-dimensional orthogonal polyhedrons. In: Advances in Automation. Proceedings of the International Russian Automation Conference, RusAutoCon 2019, 8-14 September 2019, Sochi, Russia RusAutoCon 2019. Lecture Notes in Electrical Engineering, vol. 641, pp. 891–900. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39225-3_96

  3. Shlepetinskiy, A.Y., Manzhula, K.P.: Size of a zone dangerous by damage at the root of cruciform weld joint. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 181–189. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2_16

  4. Godlevskiy, V.A., Markov, V.V., Usoltseva, N.V.: Principle of compatibility of heterogeneous additives in triboactive metalworking fluids for edge cutting of metals. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 65–71 (2017). https://doi.org/10.1007/978-3-319-53363-6_8

  5. Chekanin, V.A., Chekanin, A.V.: Packing compaction algorithm for problems of resource placement optimization. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 1–9. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2_1

  6. Chekanin, V.A., Chekanin, A.V.: Deleting objects algorithm for the optimization of orthogonal packing problems. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 27–35. Springer (2017). https://doi.org/10.1007/978-3-319-53363-6_4

  7. Tolok, A.V., Tolok, N.B.: Mathematical programming problems solving by functional voxel method. Autom. Remote Control 79(9), 1703–1712 (2018). https://doi.org/10.1134/S0005117918090138

    Article  MathSciNet  Google Scholar 

  8. Byholm, T., Toivakka, M., Westerholm, J.: Effective packing of 3-dimensional voxel-based arbitrarily shaped particles. Powder Technol. 196(2), 139–146 (2009). https://doi.org/10.1016/j.powtec.2009.07.013

    Article  Google Scholar 

  9. Masek, W.J.: Some NP-Complete Set Covering Problems: Manuscript. MIT, Cambridge (1979)

    Google Scholar 

  10. Culberson, J.C., Reckhow, R.A.: Covering polygons is hard. J. Algorithms 17(1), 2–44 (1994)

    Article  MathSciNet  Google Scholar 

  11. Khasanova, E.I., Valeev, R.S.: The matrix decomposition method of multicoherent polygon into the set of rectangular areas of minimum capacity. Vest. UGATU 14(2), 183–187 (2010). (in Russian)

    Google Scholar 

  12. Valiakhmetova, Yu.I., Telitskiy, S.V., Hasanova, E.I.: Gibridnyi algoritm na osnove posledovatel’nogo utochneniya otsenok dlya zadach maksimal’nogo ortogonal’nogo pokrytiya. Vest. Bashkirskogo Univ. 17(1), 421–424 (2012). (in Russian)

    Google Scholar 

  13. Lodi, A., Martello, S., Vigo, D.: Recent advances on two-dimensional bin packing problems. Discrete Appl. Math 123(1–3), 379–396 (2002). https://doi.org/10.1016/S0166-218X(01)00347-X

    Article  MathSciNet  MATH  Google Scholar 

  14. Floderus, P., Jansson, J., Levcopoulos, C., Lingas, A., Sledneu, D.: 3D rectangulations and geometric matrix multiplication. Algorithmica 80(1), 136–154 (2016). https://doi.org/10.1007/s00453-016-0247-3

    Article  MathSciNet  MATH  Google Scholar 

  15. Durocher, S., Mehrabi, S.: Computing conforming partitions of orthogonal polygons with minimum stabbing number. Theor. Comput. Sci. 689, 157–168 (2017). https://doi.org/10.1016/j.tcs.2017.05.035

    Article  MathSciNet  MATH  Google Scholar 

  16. Biedl, T., Derka, M., Irvine, V., Lubiw, A., Mondal, D., Turcotte, A.: Partitioning orthogonal histograms into rectangular boxes. In: Bender, M.A., Farach-Colton, M., Mosteiro, M.A. (eds.) LATIN 2018: Theoretical Informatics. Lecture Notes in Computer Science, pp. 146–160. Springer, Cham (2018)

    Chapter  Google Scholar 

  17. Cruz-Matías, I., Ayala, D.: Compact union of disjoint boxes: an efficient decomposition model for binary volumes. Comput. Sist. 21(2), 275–292 (2017)

    Google Scholar 

  18. Wu, S.Y., Sahni, S.: Fast algorithms to partition simple rectilinear polygons. VLSI Design 1(3), 193–215 (1994). https://doi.org/10.1155/1994/16075

    Article  Google Scholar 

  19. Györi, E., Hoffmann, F., Kriegel, K., Shermer, T.: Generalized guarding and partitioning for rectilinear polygons. Comput. Geomet. 6(1), 21–44 (1996). https://doi.org/10.1016/0925-7721(96)00014-4

    Article  MathSciNet  MATH  Google Scholar 

  20. Chekanin, V.A., Chekanin, A.V.: An efficient model for the orthogonal packing problem. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, vol. 22, pp. 33–38. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15684-2_5

  21. Chekanin, V.A., Chekanin, A.V.: New effective data structure for multidimensional optimization orthogonal packing problems. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 87-92. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29579-4_9

  22. Chekanin, V.A., Chekanin, A.V.: Algorithms for the formation of orthogonal polyhedrons of arbitrary dimension in the cutting and packing problems. Vest. MSTU «STANKIN» 3, 126–130 (2018). (in Russian)

    Google Scholar 

  23. Chekanin, V.A., Chekanin, A.V.: Design of library of metaheuristic algorithms for solving the problems of discrete optimization. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 25–32. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72929-9_4

Download references

Author information

Authors and Affiliations

  1. Moscow State University of Technology «STANKIN», Moscow, Russia

    Vladislav A. Chekanin & Alexander V. Chekanin

Authors
  1. Vladislav A. Chekanin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander V. Chekanin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladislav A. Chekanin .

Editor information

Editors and Affiliations

  1. Saint Petersburg Polytechnic University, Saint Petersburg, Russia

    Alexander N. Evgrafov

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chekanin, V.A., Chekanin, A.V. (2021). Solving the Problem of Decomposition of an Orthogonal Polyhedron of Arbitrary Dimension. In: Evgrafov, A.N. (eds) Advances in Mechanical Engineering. MMESE 2020. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-62062-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-62062-2_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-62061-5

  • Online ISBN: 978-3-030-62062-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation