Skip to main content
SpringerLink
Log in

On the Verification of the Correctness of a Subgraph Construction Algorithm

  • Conference paper
  • First Online:
Verification, Model Checking, and Abstract Interpretation (VMCAI 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14499))

Abstract

We automatically verify the crucial steps in the original proof of correctness of an algorithm which, given a geometric graph satisfying certain additional properties removes edges in a systematic way for producing a connected graph in which edges do not (geometrically) intersect. The challenge in this case is representing and reasoning about geometric properties of graphs in the Euclidean plane, about their vertices and edges, and about connectivity. For modelling the geometric aspects, we use an axiomatization of plane geometry; for representing the graph structure we use additional predicates; for representing certain classes of paths in geometric graphs we use linked lists.

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Notes

  1. 1.

    The other axioms, which are derived from A1, A2, A3, A4, A5, A6 are listed in the extended version of this paper [10, Appendix 6.1].

  2. 2.

    Since the clause form of some of the axioms contain more than one formula, and for doing the tests we used axioms in clause form, in such cases a range of numbers is used for the axioms.

  3. 3.

    The predicate for the vertices is used here to distinguish between the axioms for geometry, which hold for arbitrary points, and properties for edges between vertices in a graph.

  4. 4.

    This nullable subterm property has the role of excluding null pointer errors.

  5. 5.

    The tests can be found under https://github.com/sofronie/tests-vmcai-2024.git and https://userpages.uni-koblenz.de/~boeltz/CP-Algorithm-Verification/.

  6. 6.

    The tests can be found in https://github.com/sofronie/tests-vmcai-2024.git (folder Proof) and https://userpages.uni-koblenz.de/~boeltz/CP-Algorithm-Verification/Proof.

  7. 7.

    Then \(d_2=w_3x_3\) or \(d_2=x_3w_3\) or \(d_2 = w_3x_2\) or \(d_2 = x_3x_2\).

  8. 8.

    Depending on the edge intersecting with \(u_1v_1\), \(d_3\) is either \(w_3x_3\) or \(x_3w_3\).

  9. 9.

    In Step5d-f it is proven that the considered edges do not intersect with \(w_2x_2\).

  10. 10.

    A similar result is proven in tests Step 6h-j for the edges \(w_2w_4\) and \(w_3w_4\).

  11. 11.

    The tests can be found in https://github.com/sofronie/tests-vmcai-2024.git (folder Proof) and https://userpages.uni-koblenz.de/~boeltz/CP-Algorithm-Verification/Proof.

References

  1. Abdulaziz, M., Mehlhorn, K., Nipkow, T.: Trustworthy graph algorithms (invited talk). In: Rossmanith, P., Heggernes, P., Katoen, J. (eds.) 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany. LIPIcs, vol. 138, pp. 1:1–1:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.MFCS.2019.1

  2. Barriére, L., Fraigniaud, P., Narayanan, L.: Robust position-based routing in wireless ad hoc networks with unstable transmission ranges. In: Proceedings of the 5th ACM International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (DIAL M 01), New York, New York, USA, pp. 19–27. ACM Press (2001). https://doi.org/10.1145/381448.381451

  3. Beeson, M., Wos, L.: Finding proofs in Tarskian geometry. J. Autom. Reason. 58(1), 181–207 (2017). https://doi.org/10.1007/s10817-016-9392-2

    Article  MathSciNet  Google Scholar 

  4. Bettstetter, C., Hartmann, C.: Connectivity of wireless multihop networks in a shadow fading environment. Wireless Netw. 11(5), 571–579 (2005). https://doi.org/10.1007/s11276-005-3513-x

    Article  Google Scholar 

  5. Bjørner, N., de Moura, L., Nachmanson, L., Wintersteiger, C.M.: Programming Z3. In: Bowen, J.P., Liu, Z., Zhang, Z. (eds.) SETSS 2018. LNCS, vol. 11430, pp. 148–201. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17601-3_4

    Chapter  Google Scholar 

  6. Bjørner, N., Nachmanson, L.: Navigating the universe of Z3 theory solvers. In: Carvalho, G., Stolz, V. (eds.) SBMF 2020. LNCS, vol. 12475, pp. 8–24. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-63882-5_2

    Chapter  Google Scholar 

  7. Böhmer, S., Schneider, D., Frey, H.: Stochastic modeling and simulation for redundancy and coexistence in graphs resulting from log-normal shadowing. In: Proceedings of the 22nd International ACM Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems - MSWIM 2019, New York, New York, USA, pp. 173–182. ACM Press (2019). https://doi.org/10.1145/3345768.3355933

  8. Böltz, L., Becker, B., Frey, H.: Local construction of connected plane subgraphs in graphs satisfying redundancy and coexistence. In: XI Latin and American Algorithms, Graphs and Optimization Symposium (LAGOS), pp. 1–10 (2021)

    Google Scholar 

  9. Böltz, L., Frey, H.: Existence of connected intersection-free subgraphs in graphs with redundancy and coexistence property. In: Dressler, F., Scheideler, C. (eds.) ALGOSENSORS 2019. LNCS, vol. 11931, pp. 63–78. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34405-4_4

    Chapter  Google Scholar 

  10. Böltz, L., Sofronie-Stokkermans, V., Frey, H.: On the verification of the correctness of a subgraph construction algorithm (extended version), ArXiv, https://doi.org/10.48550/arXiv.2311.17860 (2023)

  11. Böltz, L., Sofronie-Stokkermans, V., Frey, H.: Tests for the verification of the CP-algorithm (2023). https://github.com/sofronie/tests-vmcai-2024.git

  12. Chen, T., Chen, J., Wu, C.: Distributed object tracking using moving trajectories in wireless sensor networks. Wirel. Networks 22(7), 2415–2437 (2016)

    Article  Google Scholar 

  13. Courcelle, B.: On the expression of monadic second-order graph properties without quantifications over sets of edges (extended abstract). In: Proceedings of the Fifth Annual Symposium on Logic in Computer Science (LICS 1990), Philadelphia, Pennsylvania, USA, June 4–7, 1990, pp. 190–196. IEEE Computer Society (1990). https://doi.org/10.1109/LICS.1990.113745

  14. Courcelle, B.: The monadic second-order logic of graphs VI: on several representations of graphs by relational structures. Discret. Appl. Math. 63(2), 199–200 (1995). https://doi.org/10.1016/0166-218X(95)00006-D

    Article  MathSciNet  Google Scholar 

  15. Courcelle, B.: The monadic second-order logic of graphs XVI: canonical graph decompositions. Log. Methods Comput. Sci. 2(2) (2006). https://doi.org/10.2168/LMCS-2(2:2)2006

  16. Courcelle, B.: Monadic second-order logic for graphs: algorithmic and language theoretical applications. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 19–22. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00982-2_2

    Chapter  Google Scholar 

  17. Cox, D.A., Little, J., O’Shea, D.: Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.). Undergraduate texts in mathematics, Springer (1997)

    Google Scholar 

  18. Doczkal, C., Pous, D.: Graph theory in Coq: minors, treewidth, and isomorphisms. J. Autom. Reason. 64, 795–825 (2020)

    Article  MathSciNet  Google Scholar 

  19. Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. J. Autom. Reason. 21(3), 357–380 (1998). https://doi.org/10.1023/A:1006031329384

    Article  MathSciNet  Google Scholar 

  20. Frey, H., Simplot-Ryl, D.: Localized topology control algorithms for ad hoc and sensor networks. In: Nayak, A., Stojmenovic, I. (eds.) Handbook of Applied Algorithms, chap. 15, pp. 439–464. Wiley, Hoboken (2007)

    Google Scholar 

  21. Frey, H., Stojmenovic, I.: On delivery guarantees and worst-case forwarding bounds of elementary face routing components in ad hoc and sensor networks. IEEE Trans. Comput. 59(9), 1224–1238 (2010). https://doi.org/10.1109/TC.2010.107

    Article  MathSciNet  Google Scholar 

  22. Heckel, R., Lambers, L., Saadat, M.G.: Analysis of graph transformation systems: Native vs translation-based techniques. In: Electronic Proceedings in Theoretical Computer Science, vol. 309, pp. 1–22 (2019). https://doi.org/10.4204/eptcs.309.1

  23. Hilbert, D.: The Foundations of Geometry. 2nd ed. Chicago: Open Court. (1980 (1899))

    Google Scholar 

  24. Ihlemann, C., Jacobs, S., Sofronie-Stokkermans, V.: On local reasoning in verification. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 265–281. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_19

    Chapter  Google Scholar 

  25. Ihlemann, C., Sofronie-Stokkermans, V.: System description: H-PILoT. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 131–139. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02959-2_9

    Chapter  Google Scholar 

  26. Karamete, B.K., Adhami, L., Glaser, E.: A fixed storage distributed graph database hybrid with at-scale OLAP expression and I/O support of a relational DB: kinetica-graph. CoRR abs/2201.02136 (2022). https://arxiv.org/abs/2201.02136

  27. Kuhn, F., Wattenhofer, R., Zollinger, A.: Ad-hoc networks beyond unit disk graphs. In: ACM DIALM-POMC Joint Workshop on Foundations of Mobile Computing, pp. 69–78. San Diego (2003)

    Google Scholar 

  28. Mathews, E.: Planarization of geographic cluster-based overlay graphs in realistic wireless networks. In: 2012 Ninth International Conference on Information Technology - New Generations, pp. 95–101. IEEE (2012)

    Google Scholar 

  29. Mathews, E., Frey, H.: A localized link removal and addition based planarization algorithm. In: Bononi, L., Datta, A.K., Devismes, S., Misra, A. (eds.) ICDCN 2012. LNCS, vol. 7129, pp. 337–350. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-25959-3_25

    Chapter  Google Scholar 

  30. McPeak, S., Necula, G.C.: Data structure specifications via local equality axioms. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 476–490. Springer, Heidelberg (2005). https://doi.org/10.1007/11513988_47

    Chapter  Google Scholar 

  31. Mirzaie, M., Mazinani, S.M.: MCFL: an energy efficient multi-clustering algorithm using fuzzy logic in wireless sensor network. Wirel. Networks 24(6), 2251–2266 (2018)

    Article  Google Scholar 

  32. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

    Chapter  Google Scholar 

  33. Neumann, F., Vivas Estevao, D., Ockenfeld, F., Radak, J., Frey, H.: Short paper: structural network properties for local planarization of wireless sensor networks. In: Mitton, N., Loscri, V., Mouradian, A. (eds.) ADHOC-NOW 2016. LNCS, vol. 9724, pp. 229–233. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40509-4_16

    Chapter  Google Scholar 

  34. Peuter, D., Sofronie-Stokkermans, V.: On invariant synthesis for parametric systems. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 385–405. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29436-6_23

    Chapter  Google Scholar 

  35. Philip, S.J., Ghosh, J., Ngo, H.Q., Qiao, C.: Routing on overlay graphs in mobile ad hoc networks. In: Proceedings of the IEEE Global Communications Conference, Exhibition & Industry Forum (GLOBECOM’06) (2006)

    Google Scholar 

  36. Piskac, R., de Moura, L.M., Bjørner, N.S.: Deciding effectively propositional logic using DPLL and substitution sets. J. Autom. Reason. 44(4), 401–424 (2010). https://doi.org/10.1007/s10817-009-9161-6

  37. Sofronie-Stokkermans, V.: Hierarchic reasoning in local theory extensions. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 219–234. Springer, Heidelberg (2005). https://doi.org/10.1007/11532231_16

    Chapter  Google Scholar 

  38. Sturm, T., Weispfenning, V.: Computational geometry problems in REDLOG. In: Wang, D. (ed.) ADG 1996. LNCS, vol. 1360, pp. 58–86. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0022720

    Chapter  Google Scholar 

  39. Tarski, A., Givant, S.: Tarski’s system of geometry. Bull. Symb. Log. 5(2), 175–214 (1999). https://doi.org/10.2307/421089

    Article  MathSciNet  Google Scholar 

  40. Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS Version 3.5. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 140–145. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02959-2_10

    Chapter  Google Scholar 

Download references

Acknowledgment

We thank the reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. University of Koblenz, Koblenz, Germany

    Lucas Böltz, Viorica Sofronie-Stokkermans & Hannes Frey

Authors
  1. Lucas Böltz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viorica Sofronie-Stokkermans
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hannes Frey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lucas Böltz .

Editor information

Editors and Affiliations

  1. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Rayna Dimitrova

  2. Tel Aviv University, Tel Aviv, Israel

    Ori Lahav

  3. New York University, New York, NY, USA

    Sebastian Wolff

Rights and permissions

Reprints and permissions

Copyright information

© 2024 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

Böltz, L., Sofronie-Stokkermans, V., Frey, H. (2024). On the Verification of the Correctness of a Subgraph Construction Algorithm. In: Dimitrova, R., Lahav, O., Wolff, S. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2024. Lecture Notes in Computer Science, vol 14499. Springer, Cham. https://doi.org/10.1007/978-3-031-50524-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-50524-9_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-50523-2

  • Online ISBN: 978-3-031-50524-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation