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Insights into (k, ρ)-Shortcutting Algorithms

Authors Alexander Leonhardt , Ulrich Meyer , Manuel Penschuck

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

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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

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