Cart (Loading....) | Create Account
Close category search window
 

Every decision tree has an influential variable

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

4 Author(s)
O'Donnell, R. ; Microsoft Res., Redmond, VT, USA ; Saks, M. ; Schramm, O. ; Servedio, R.A.

We prove that for any decision tree calculating a Boolean function f : {-1,1}n → {-1, 1}, Var[f] ≤ Σ i=1 n δiInfi(f), i = 1 where δi is the probability that the ith input variable is read and Infi(f) is the influence of the ith variable on f. The variance, influence and probability are taken with respect to an arbitrary product measure on {-1, 1}nn. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced Boolean function with a decision tree of depth d has a variable with influence at least 1/d. The only previous nontrivial lower bound known was Ω(d2-d). Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with nonBoolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at leastΩ(v43//p13/), where v is the number of vertices and p ≤ 1/2 is the critical threshold probability. This supersedes the milestone Ω(v43//p13/) bound of Hajnal (1991) and is sometimes superior to the best known lower bounds of Chakrabarti-Khot (2001) and Friedgut-Kahn-Wigderson (2002).

Published in:

Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on

Date of Conference:

23-25 Oct. 2005

Need Help?


IEEE Advancing Technology for Humanity About IEEE Xplore | Contact | Help | Terms of Use | Nondiscrimination Policy | Site Map | Privacy & Opting Out of Cookies

A not-for-profit organization, IEEE is the world's largest professional association for the advancement of technology.
© Copyright 2014 IEEE - All rights reserved. Use of this web site signifies your agreement to the terms and conditions.