  In the communication problem $\mathbf{UR}$ (universal relation) [KRW95],
Alice and Bob respectively receive $x, y \in\{0,1\}^n$ with the promise that
$x\neq y$. The last player to receive a message must output an index $i$ such
that $x_i\neq y_i$. We prove that the randomized one-way communication
complexity of this problem in the public coin model is exactly
$\Theta(\min\{n,\log(1/\delta)\log^2(\frac n{\log(1/\delta)})\})$ for failure
probability $\delta$. Our lower bound holds even if promised
$\mathop{support}(y)\subset \mathop{support}(x)$. As a corollary, we obtain
optimal lower bounds for $\ell_p$-sampling in strict turnstile streams for
$0\le p < 2$, 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
$x\in\{0,1\}^n$ at all points in the stream.
  We give two different proofs of our main result. The first proof demonstrates
that any algorithm $\mathcal A$ 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 $\mathcal A$ 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 $\mathcal A$, which is loosely
motivated by techniques in differential privacy. Our second proof is via a
novel randomized reduction from Augmented Indexing [MNSW98] which needs to
interact with $\mathcal A$ 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 $\mathcal A$ in the
reduction.
