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,
EditorsTiziana Margaria-Steffen, Giuseppe F. Italiano, Jean-Jacques Quisquater, Roger Wattenhofer, Jaroslav Pokorný
PublisherSpringer Verlag
Pages412-422
Number of pages11
ISBN (Electronic)9783662460771
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

Fingerprint

Decision trees
Threshold Circuits
Decision tree
Lower bound
Threshold Function
Boolean functions
Networks (circuits)
Boolean Functions
Absolute value
Scalar, inner or dot product
Linear Function
Modulo
Leaves
Internal
Integer
Computing
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Uchizawa, K., & Takimoto, E. (2015). Lower bounds for linear decision trees with bounded weights. In T. Margaria-Steffen, G. F. Italiano, J-J. Quisquater, R. Wattenhofer, & J. Pokorný (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.

Lower bounds for linear decision trees with bounded weights. / Uchizawa, Kei; Takimoto, Eiji.

SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,. ed. / Tiziana Margaria-Steffen; Giuseppe F. Italiano; Jean-Jacques Quisquater; Roger Wattenhofer; Jaroslav Pokorný. Springer Verlag, 2015. p. 412-422 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8939).

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

Uchizawa, K & Takimoto, E 2015, Lower bounds for linear decision trees with bounded weights. in T Margaria-Steffen, GF Italiano, J-J Quisquater, R Wattenhofer & J Pokorný (eds), SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8939, Springer Verlag, pp. 412-422, 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015, Pec pod Sněžkou, Czech Republic, 1/24/15.
Uchizawa K, Takimoto E. Lower bounds for linear decision trees with bounded weights. In Margaria-Steffen T, Italiano GF, Quisquater J-J, Wattenhofer R, Pokorný J, editors, SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,. Springer Verlag. 2015. p. 412-422. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Uchizawa, Kei ; Takimoto, Eiji. / Lower bounds for linear decision trees with bounded weights. SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,. editor / Tiziana Margaria-Steffen ; Giuseppe F. Italiano ; Jean-Jacques Quisquater ; Roger Wattenhofer ; Jaroslav Pokorný. Springer Verlag, 2015. pp. 412-422 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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