Lower bounds for linear decision trees with bounded weights

Kei Uchizawa, Eiji Takimoto

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w+1)r threshold gates with weight at most (2w+1)r+1 in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2Ω(n/ logw) lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound 2√n if w = 2o(√n).

Original languageEnglish
Title of host publicationSOFSEM 2015
Subtitle of host publicationTheory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,
EditorsGiuseppe F. Italiano, Tiziana Margaria-Steffen, Jaroslav Pokorný, Jean-Jacques Quisquater, Roger Wattenhofer
PublisherSpringer Verlag
Pages412-422
Number of pages11
ISBN (Electronic)9783662460771
DOIs
Publication statusPublished - Jan 1 2015
Event41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015 - Pec pod Sněžkou, Czech Republic
Duration: Jan 24 2015Jan 29 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8939
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015
CountryCzech Republic
CityPec pod Sněžkou
Period1/24/151/29/15

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Uchizawa, K., & Takimoto, E. (2015). Lower bounds for linear decision trees with bounded weights. In G. F. Italiano, T. Margaria-Steffen, J. Pokorný, J-J. Quisquater, & R. Wattenhofer (Eds.), SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science, (pp. 412-422). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8939). Springer Verlag. https://doi.org/10.1007/978-3-662-46078-8_34