Lower bounds for linear decision trees via an energy complexity argument

Kei Uchizawa, Eiji Takimoto

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

4 Citations (Scopus)

Abstract

A linear decision tree is a binary decision tree in which a classification rule at each internal node is defined by a linear threshold function. In this paper, we consider a linear decision tree T where the weights w1, w2, ..., wn of each linear threshold function satisfy ∑i|wi| ≤ w for an integer w, and prove that if T computes an n-variable Boolean function of large unbounded-error communication complexity (such as the Inner-Product function modulo two), then T must have 2Ω(√n)/w leaves. To obtain the lower bound, we utilize a close relationship between the size of linear decision trees and the energy complexity of threshold circuits; the energy of a threshold circuit C is defined to be the maximum number of gates outputting "1," where the maximum is taken over all inputs to C. In addition, we consider threshold circuits of depth ω(1) and bounded energy, and provide two exponential lower bounds on the size (i.e., the number of gates) of such circuits.

Original languageEnglish
Title of host publicationMathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings
Pages568-579
Number of pages12
DOIs
Publication statusPublished - Sep 1 2011
Event36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011 - Warsaw, Poland
Duration: Aug 22 2011Aug 26 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6907 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011
CountryPoland
CityWarsaw
Period8/22/118/26/11

Fingerprint

Threshold Circuits
Decision trees
Decision tree
Lower bound
Threshold Function
Networks (circuits)
Energy
Linear Function
Boolean functions
Classification Rules
Communication Complexity
Boolean Functions
Scalar, inner or dot product
Modulo
Leaves
Binary
Internal
Integer
Communication
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Uchizawa, K., & Takimoto, E. (2011). Lower bounds for linear decision trees via an energy complexity argument. In Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings (pp. 568-579). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6907 LNCS). https://doi.org/10.1007/978-3-642-22993-0_51

Lower bounds for linear decision trees via an energy complexity argument. / Uchizawa, Kei; Takimoto, Eiji.

Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings. 2011. p. 568-579 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6907 LNCS).

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

Uchizawa, K & Takimoto, E 2011, Lower bounds for linear decision trees via an energy complexity argument. in Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6907 LNCS, pp. 568-579, 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011, Warsaw, Poland, 8/22/11. https://doi.org/10.1007/978-3-642-22993-0_51
Uchizawa K, Takimoto E. Lower bounds for linear decision trees via an energy complexity argument. In Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings. 2011. p. 568-579. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-22993-0_51
Uchizawa, Kei ; Takimoto, Eiji. / Lower bounds for linear decision trees via an energy complexity argument. Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings. 2011. pp. 568-579 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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