TY - JOUR
T1 - An optimal tester for k-linear
AU - Bshouty, Nader H.
N1 - Publisher Copyright:
© 2023 Elsevier B.V.
PY - 2023/3/16
Y1 - 2023/3/16
N2 - A Boolean function f:{0,1}n→{0,1} is k-linear if it returns the sum (over the binary field F2) of exactly k coordinates of the input. In this paper, we study property testing of the classes k-Linear, the class of all k-linear functions, and k-Linear⁎, the class ∪j=0 kj-Linear. We give a non-adaptive distribution-free two-sided ϵ-tester for k-Linear that makes [Formula presented] queries. This matches the lower bound known from the literature. We then give a non-adaptive distribution-free one-sided ϵ-tester for k-Linear⁎ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)logn+Ω(1/ϵ) queries. The latter bound almost matches the upper bound O(klogn+1/ϵ) known from the literature. We then show that any adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)logn+Ω(1/ϵ) queries.
AB - A Boolean function f:{0,1}n→{0,1} is k-linear if it returns the sum (over the binary field F2) of exactly k coordinates of the input. In this paper, we study property testing of the classes k-Linear, the class of all k-linear functions, and k-Linear⁎, the class ∪j=0 kj-Linear. We give a non-adaptive distribution-free two-sided ϵ-tester for k-Linear that makes [Formula presented] queries. This matches the lower bound known from the literature. We then give a non-adaptive distribution-free one-sided ϵ-tester for k-Linear⁎ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)logn+Ω(1/ϵ) queries. The latter bound almost matches the upper bound O(klogn+1/ϵ) known from the literature. We then show that any adaptive uniform-distribution one-sided ϵ-tester for k-Linear must make at least Ω˜(k)logn+Ω(1/ϵ) queries.
KW - Linear functions
KW - Property testing
UR - http://www.scopus.com/inward/record.url?scp=85150809521&partnerID=8YFLogxK
U2 - 10.1016/j.tcs.2023.113759
DO - 10.1016/j.tcs.2023.113759
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AN - SCOPUS:85150809521
SN - 0304-3975
VL - 950
JO - Theoretical Computer Science
JF - Theoretical Computer Science
M1 - 113759
ER -