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Fragile Complexity of Comparison-Based Algorithms

Authors Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, Nodari Sitchinava

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Algorithms
  • comparison based algorithms
  • lower bounds

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Abstract

We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible.

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Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava. Fragile Complexity of Comparison-Based Algorithms. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.2

Author Details

Peyman Afshani
  • Aarhus University, Denmark
Rolf Fagerberg
  • University of Southern Denmark, Odense, Denmark
David Hammer
  • Goethe University Frankfurt, Germany
  • University of Southern Denmark, Odense, Denmark
Riko Jacob
  • IT University of Copenhagen, Denmark
Irina Kostitsyna
  • TU Eindhoven, The Netherlands
Ulrich Meyer
  • Goethe University Frankfurt, Germany
Manuel Penschuck
  • Goethe University Frankfurt, Germany
Nodari Sitchinava
  • University of Hawaii at Manoa

Funding

  • Fagerberg, Rolf: Supported by the Independent Research Fund Denmark, Natural Sciences, grant DFF-7014-00041.
  • Hammer, David: Supported by the Deutsche Forschungsgemeinschaft (DFG) under grants ME 2088/3-2 and ME 2088/4-2.
  • Meyer, Ulrich: Supported by the Deutsche Forschungsgemeinschaft (DFG) under grants ME 2088/3-2 and ME 2088/4-2.
  • Penschuck, Manuel: Supported by the Deutsche Forschungsgemeinschaft (DFG) under grants ME 2088/3-2 and ME 2088/4-2.
  • Sitchinava, Nodari: Supported by the National Science Foundation under Grant No. CCF-1533823.

Acknowledgements

We thank Steven Skiena for posing the original problem, and we thank Michael Bender, Rob Johnson, and Pat Morin for helpful discussions.

References

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