TY - JOUR

T1 - Testing juntas

AU - Fischer, Eldar

AU - Kindler, Guy

AU - Ron, Dana

AU - Safra, Shmuel

AU - Samorodnitsky, Alex

N1 - Funding Information:
·Corresponding author. E-mail address: eldar@cs.technion.ac.il (E. Fischer). 1Research supported by a Technion VPR fund Dent Charitable Trust, non-military research fund, and by a joint Haifa University–Technion research fund. 2Research supported by the Israel Science Foundation (Grant 32/00-1). 3Research supported by an Israeli Science Foundation grant and a United States–Israel Binational Science Foundation grant. 4Research supported by the Israel Science Foundation (Grant 039-7165).

PY - 2004/6

Y1 - 2004/6

N2 - We show that a boolean valued function over n variables, where each variable ranges in an arbitrary probability space, can be tested for the property of depending on only J of them using a number of queries that depends only polynomially on J and the approximation parameter ε. We present several tests that require a number of queries that is polynomial in J and linear in ε-1. We show a non-adaptive test that has one-sided error, an adaptive version of it that requires fewer queries, and a non-adaptive two-sided version of the test that requires the least number of queries. We also show a two-sided non-adaptive test that applies to functions over n boolean variables, and has a more compact analysis. We then provide a lower bound of Ω̃(J) on the number of queries required for the non-adaptive testing of the above property; a lower bound of Ω(log(J+1)) for adaptive algorithms naturally follows from this. In establishing this lower bound we also prove a result about random walks on the group Zq2 that may be interesting in its own right. We show that for some t(q)=Õ(q 2), the distributions of the random walk at times t and t+2 are close to each other, independently of the step distribution of the walk. We also discuss related questions. In Particular, when given in advance a known J-junta function h, we show how to test a function for the property of being identical to h up to a permutation of the variables, in a number of queries that is polynomial in J and ε-1.

AB - We show that a boolean valued function over n variables, where each variable ranges in an arbitrary probability space, can be tested for the property of depending on only J of them using a number of queries that depends only polynomially on J and the approximation parameter ε. We present several tests that require a number of queries that is polynomial in J and linear in ε-1. We show a non-adaptive test that has one-sided error, an adaptive version of it that requires fewer queries, and a non-adaptive two-sided version of the test that requires the least number of queries. We also show a two-sided non-adaptive test that applies to functions over n boolean variables, and has a more compact analysis. We then provide a lower bound of Ω̃(J) on the number of queries required for the non-adaptive testing of the above property; a lower bound of Ω(log(J+1)) for adaptive algorithms naturally follows from this. In establishing this lower bound we also prove a result about random walks on the group Zq2 that may be interesting in its own right. We show that for some t(q)=Õ(q 2), the distributions of the random walk at times t and t+2 are close to each other, independently of the step distribution of the walk. We also discuss related questions. In Particular, when given in advance a known J-junta function h, we show how to test a function for the property of being identical to h up to a permutation of the variables, in a number of queries that is polynomial in J and ε-1.

KW - Boolean functions

KW - Discrete Fourier Analysis

KW - Juntas

KW - Property testing

UR - http://www.scopus.com/inward/record.url?scp=3042662710&partnerID=8YFLogxK

U2 - 10.1016/j.jcss.2003.11.004

DO - 10.1016/j.jcss.2003.11.004

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AN - SCOPUS:3042662710

VL - 68

SP - 753

EP - 787

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 4

ER -