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Computing the Maximum using (min, +) Formulas

Authors Meena Mahajan, Prajakta Nimbhorkar, Anuj Tawari

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Meena Mahajan
Prajakta Nimbhorkar
Anuj Tawari

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Meena Mahajan, Prajakta Nimbhorkar, and Anuj Tawari. Computing the Maximum using (min, +) Formulas. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 74:1-74:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.74

Abstract

We study computation by formulas over (min,+). We consider the computation of max{x_1,...,x_n} over N as a difference of (min,+) formulas, and show that size n + n \log n is sufficient and necessary. Our proof also shows that any (min,+) formula computing the minimum of all sums of n-1 out of n variables must have n \log n leaves; this too is tight. Our proofs use a complexity measure for (min,+) functions based on minterm-like behaviour and on the entropy of an associated graph.
Keywords
  • Formulas
  • Circuits
  • Lower bounds
  • Tropical semiring

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