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• W2964156975 abstract "In the communication problem UR (universal relation) [25], Alice and Bob respectively receive x, y ∈ {0, 1} <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sup> with the promise that x ≠ y. The last player to receive a message must output an index i such that x <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</sub> ≠ y <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</sub> . We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly Θ(min{n, log(1/δ) log <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> (n/log(1/δ) )}) for failure probability δ. Our lower bound holds even if promised support(y) ⊂ support(x). As a corollary, we obtain optimal lower bounds for ℓ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</sub> -sampling in strict turnstile streams for 0 ≤ p <; 2, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised x ∈ {0, 1} <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sup> at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm A solving sampling problems in turnstile streams in low memory can be used to encode subsets of [n] of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to A throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with A, which is loosely motivated by techniques in differential privacy. Our correctness analysis involves understanding the ability of A to correctly answer adaptive queries which have positive but bounded mutual information with A's internal randomness, and may be of independent interest in the newly emerging area of adaptive data analysis with a theoretical computer science lens. Our second proof is via a novel randomized reduction from Augmented Indexing [30] which needs to interact with A adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from A in the reduction." @default.
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• W2964156975 date "2017-10-01" @default.
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• W2964156975 title "Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams" @default.
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