Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams

In the communication problem UR (universal relation), Alice and Bob respectively receive x, y ∊{0,1\}^n with the promise that x≠ y. The last player to receive a message must output an index i such that x_i≠ y_i. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly \Theta(\min\{n,\log(1/δ)\log^2(\frac n{\log(1/δ)})\}) for failure probability δ. Our lower bound holds even if promised \mathop{support}(y)⊄ \mathop{support}(x). As a corollary, we obtain optimal lower bounds for ℓ_p-sampling in strict turnstile streams for 0\le p streams for 0 ≤ p