Document Open Access Logo

Insights into (k, ρ)-Shortcutting Algorithms

Authors Alexander Leonhardt , Ulrich Meyer , Manuel Penschuck

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.84.pdf
  • Filesize: 3.16 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Complexity
  • Approximation
  • Optimal algorithms
  • Parallel shortest path

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A graph is called a (k, ρ)-graph iff every node can reach ρ of its nearest neighbors in at most k hops. This property has proven useful in the analysis and design of parallel shortest-path algorithms [Blelloch et al., 2016; Dong et al., 2021]. Any graph can be transformed into a (k, ρ)-graph by adding shortcuts. Formally, the (k,ρ)-Minimum-Shortcut-Problem (kρ-MSP) asks to find an appropriate shortcut set of minimal cardinality.
We show that kρ-MSP is NP-complete in the practical regime of k ≥ 3 and ρ = Θ(n^ε) for ε > 0. With a related construction, we bound the approximation factor of known kρ-MSP heuristics [Blelloch et al., 2016] from below and propose algorithmic countermeasures improving the approximation quality. Further, we describe an integer linear problem (ILP) that optimally solves kρ-MSP. Finally, we compare the practical performance and quality of all algorithms empirically.

Cite As Get BibTex

Alexander Leonhardt, Ulrich Meyer, and Manuel Penschuck. Insights into (k, ρ)-Shortcutting Algorithms. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.84

Author Details

Alexander Leonhardt
  • Goethe University Frankfurt, Germany
Ulrich Meyer
  • Goethe University Frankfurt, Germany
Manuel Penschuck
  • Goethe University Frankfurt, Germany

Funding

  • Penschuck, Manuel: Funded by the Deutsche Forschungsgemeinschaft (DFG) - {ME 2088/5-2} (FOR 2975 - Algorithms, Dynamics, and Information Flow in Networks).

Acknowledgements

The authors thank the anonymous reviewers for their insightful comments which greatly improved this paper.

Supplementary Materials

References

  1. Ittai Abraham, Daniel Delling, Andrew V. Goldberg, and Renato F. Werneck. Hierarchical hub labelings for shortest paths. In Algorithms - ESA 2012, pages 24-35, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  2. Haris Angelidakis, Yury Makarychev, and Vsevolod Oparin. Algorithmic and hardness results for the hub labeling problem. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 1442-1461, USA, 2017. Society for Industrial and Applied Mathematics. Google Scholar
  3. Maxim Babenko, Andrew V. Goldberg, Haim Kaplan, Ruslan Savchenko, and Mathias Weller. On the complexity of hub labeling (extended abstract). In Mathematical Foundations of Computer Science 2015, pages 62-74, Berlin, Heidelberg, 2015. Springer Berlin Heidelberg. Google Scholar
  4. Albert-László Barabási. Network science book. Network Science, 625, 2014. Google Scholar
  5. Richard Bellman. On a routing problem. Quarterly of Applied Mathematics, 16(1):87-90, 1958. URL: http://www.jstor.org/stable/43634538.
  6. Shankar Bhamidi, Remco Van der Hofstad, and Gerard Hooghiemstra. First passage percolation on the erdős-rényi random graph. Combinatorics, Probability and Computing, 20(5):683-707, 2011. Google Scholar
  7. Guy E. Blelloch, Yan Gu, Yihan Sun, and Kanat Tangwongsan. Parallel Shortest Paths Using Radius Stepping. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '16, pages 443-454, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2935764.2935765.
  8. Andrei Z. Broder. On the Resemblance and Containment of Documents. In Proceedings of the Compression and Complexity of Sequences 1997, SEQUENCES '97, page 21, USA, 1997. IEEE Computer Society. Google Scholar
  9. Edith Cohen, Eran Halperin, Haim Kaplan, and Uri Zwick. Reachability and distance queries via 2-hop labels. SIAM Journal on Computing, 32(5):1338-1355, 2003. URL: https://doi.org/10.1137/S0097539702403098.
  10. Manuel Costanzo, Enzo Rucci, Marcelo Naiouf, and Armando De Giusti. Performance vs Programming Effort between Rust and C on Multicore Architectures: Case Study in N-Body. In 2021 XLVII Latin American Computing Conference (CLEI), pages 1-10, 2021. URL: https://doi.org/10.1109/CLEI53233.2021.9640225.
  11. Daniel Delling, Andrew V. Goldberg, Ruslan Savchenko, and Renato F. Werneck. Hub labels: Theory and practice. In Proceedings of the 13th International Symposium on Experimental Algorithms - Volume 8504, pages 259-270, Berlin, Heidelberg, 2014. Springer-Verlag. URL: https://doi.org/10.1007/978-3-319-07959-2_22.
  12. Edsger W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1(1):269-271, 1959. Google Scholar
  13. Xiaojun Dong, Yan Gu, Yihan Sun, and Yunming Zhang. Efficient stepping algorithms and implementations for parallel shortest paths. In Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures, pages 184-197, 2021. Google Scholar
  14. Tobias Friedrich and Anton Krohmer. On the diameter of hyperbolic random graphs. SIAM Journal on Discrete Mathematics, 32(2):1314-1334, 2018. URL: https://doi.org/10.1137/17M1123961.
  15. Edgar N. Gilbert. Random Graphs. The Annals of Mathematical Statistics, 30:1141-1144, 1959. URL: http://www.jstor.org/stable/2237458.
  16. Christos Gkantsidis, Milena Mihail, and Ellen W. Zegura. The Markov Chain Simulation Method for Generating Connected Power Law Random Graphs. In Proceedings of the Fifth Workshop on Algorithm Engineering and Experiments, Baltimore, MD, USA, January 11, 2003, pages 16-25. SIAM, 2003. Google Scholar
  17. Siddharth Gupta, Adrian Kosowski, and Laurent Viennot. Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132, pages 143:1-143:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.143.
  18. Lester R. Ford Jr. Network Flow Theory. RAND Corporation, Santa Monica, CA, 1956. Google Scholar
  19. Richard M. Karp. Reducibility Among Combinatorial Problems. In Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. Google Scholar
  20. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguñá. Hyperbolic geometry of complex networks. Phys. Rev. E, 82:036106, September 2010. URL: https://doi.org/10.1103/PhysRevE.82.036106.
  21. Alexander Leonhardt, Ulrich Meyer, and Manuel Penschuck. K-Rho-Shortcutting Heuristics. Software, (visited on 2024-08-09). URL: https://github.com/alleonhardt/k-rho-shortcutting
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.22475
  22. Alexander Leonhardt, Ulrich Meyer, and Manuel Penschuck. Insights into (k,ρ)-shortcutting algorithms, 2024. URL: https://arxiv.org/abs/2402.07771.
  23. Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014.
  24. Ulrich Meyer and Peter Sanders. Δ-Stepping: A Parallel Single Source Shortest Path Algorithm. In Algorithms - ESA' 98, pages 393-404, Berlin, Heidelberg, 1998. Springer Berlin Heidelberg. Google Scholar
  25. Mathew Penrose. Random Geometric Graphs. Oxford University Press, May 2003. URL: https://doi.org/10.1093/acprof:oso/9780198506263.001.0001.
  26. Ryan A. Rossi and Nesreen K. Ahmed. The network data repository with interactive graph analytics and visualization. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015. URL: http://networkrepository.com.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact