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Fast Convolutions for Near-Convex Sequences

Authors Cornelius Brand , Alexandra Lassota

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • (min,+)-convolution
  • fine-grained complexity
  • convex sequences

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Abstract

We develop algorithms for (min,+)-Convolution and related convolution problems such as Super Additivity Testing, Convolution 3-Sum and Minimum Consecutive Subsums which use the degree of convexity of the instance as a parameter. Assuming the min-plus conjecture (Künnemann-Paturi-Schneider, ICALP'17 and Cygan et al., ICALP'17), our results interpolate in an optimal manner between fully convex instances, which can be solved in near-linear time using Legendre transformations, and general non-convex sequences, where the trivial quadratic-time algorithm is conjectured to be best possible, up to subpolynomial factors.

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Cornelius Brand and Alexandra Lassota. Fast Convolutions for Near-Convex Sequences. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.16

Author Details

Cornelius Brand
  • Institute of Logic and Computation, Vienna University of Technology, Austria
Alexandra Lassota
  • Max Planck Institute for Informatics, SIC, Saarbrücken, Germany

Funding

The research leading to this article was partially carried out at EPFL, Lausanne, Switzerland, funded by the Swiss National Science Foundation project Complexity of integer Programming (207365).
  • Brand, Cornelius: Supported by FWF grant Y1329 in the START-Program (ParAI).
  • Lassota, Alexandra: Funded by the Swiss National Science Foundation project Complexity of integer Programming (207365).

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