Abstract-Let X and Y be finite non-empty sets and (X, Y ) a pair of random variables taking values in X × Y. We consider communication protocols between two parties, Alice and Bob, for generating X and Y . Alice is provided an x ∈ X generated according to the distribution of X, and is required to send a message to Bob in order to enable him to generate y ∈ Y, whose distribution is the same as that of Y |X=x. Both parties have access to a shared random string generated in advance. Let T (X : Y ) be the minimum (over all protocols) of the expected number of bits Alice needs to transmit to achieve this. We show thatWe also consider the worst-case communication required for this problem, where we seek to minimize the average number of bits Alice must transmit for the worst-case x ∈ X . We show that the communication required in this case is related to the capacity C(E) of the channel E, derived from (X, Y ), that maps x ∈ X to the distribution of Y |X=x. We show that the required communication T (E) satisfiesUsing the first result, we derive a direct sum theorem in communication complexity that substantially improves the previous such result shown by Jain, Radhakrishnan and Sen [In Proc. 30th International Colloquium of Automata, Languages and Programming (ICALP), ser. LNCS, vol. 2719LNCS, vol. . 2003.These results are obtained by employing a rejection sampling procedure that relates the relative entropy between two distributions to the communication complexity of generating one distribution from the other.
Abstract-Let X and Y be finite non-empty sets and (X, Y ) a pair of random variables taking values in X × Y. We consider communication protocols between two parties, Alice and Bob, for generating X and Y . Alice is provided an x ∈ X generated according to the distribution of X, and is required to send a message to Bob in order to enable him to generate y ∈ Y, whose distribution is the same as that of Y |X=x. Both parties have access to a shared random string generated in advance. Let T (X : Y ) be the minimum (over all protocols) of the expected number of bits Alice needs to transmit to achieve this. We show thatWe also consider the worst-case communication required for this problem, where we seek to minimize the average number of bits Alice must transmit for the worst-case x ∈ X . We show that the communication required in this case is related to the capacity C(E) of the channel E, derived from (X, Y ), that maps x ∈ X to the distribution of Y |X=x. We show that the required communication T (E) satisfiesUsing the first result, we derive a direct sum theorem in communication complexity that substantially improves the previous such result shown by Jain, Radhakrishnan and Sen [In Proc. 30th International Colloquium of Automata, Languages and Programming (ICALP), ser. LNCS, vol. 2719LNCS, vol. . 2003.These results are obtained by employing a rejection sampling procedure that relates the relative entropy between two distributions to the communication complexity of generating one distribution from the other.
For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of n-bit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires Ω(n) queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with O(√ n) queries [E. Fischer et al., Monotonicity testing over general poset domains, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 474-483]. Notice that for every negative instance (i.e., an assignment that does not satisfy ϕ) there are three bit queries that witness this fact. Nevertheless, finding such a short witness requires reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property. A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include the following observations which are of independent interest: 1. In the context of testing for linear properties, adaptive two-sided error tests have no more power than nonadaptive one-sided error tests. Moreover, without loss of generality, any test for a linear property is a linear test. A linear test verifies that a portion of the input satisfies a set of linear constraints, which define the property, and rejects if and only if it finds a falsified constraint. A linear test is by definition nonadaptive and, when applied to linear properties, has a one-sided error. 2. Random low density parity check codes (which are known to have linear distance and constant rate) are not locally testable. In fact, testing such a code of length n requires Ω(n) queries.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.