Two Player Hidden Pointer Chasing and Multi-Pass Lower Bounds in Turnstile Streams

Abstract: The authors have withdrawn this paper due to an error in the proof of Lemma 3.4. -------------------------------------------------------------------------------------------- The authors have withdrawn this paper due to an error in the proof of Lemma 3.4z(Assadi, Chen, and Khanna, 2019) define a 4-player hidden-pointer-chasing ($\mathsf{HPC}<sup>4$), and using it, give strong multi-pass lower bounds for graph problems in the streaming model of computation and a lower bound on the query complexity of sub-modular minimization. We present a two-player version ($\mathsf{HPC}<sup>2$) of $\mathsf{HPC}<sup>4$ that has matching communication complexity to $\mathsf{HPC}<sup>4$. Our formulation allows us to lower bound its communication complexity with a simple direct-sum argument. Using this lower bound on the communication complexity of $\mathsf{HPC}<sup>2$, we retain the streaming and query complexity lower bounds by (Assadi, Chen, and Khanna, 2019). Further, by giving reductions from $\mathsf{HPC}<sup>2$, we prove new multi-pass space lower bounds for graph problems in turnstile streams. In particular, we show that any algorithm which computes the exact weight of the maximum weighted matching in an $n$-vertex graph requires $\tilde{O}(n<sup>{2})$ space unless it makes $\omega(\log n)$ passes over the turnstile stream, and that any algorithm which computes the minimum $s\text{-}t$ distance in an $n$-vertex graph requires $n<sup>{2-o(1)}$ space unless it makes $n<sup>{\Omega(1)}$ passes over the turnstile stream. Our reductions can be modified to use $\mathsf{HPC}<sup>4$ as well.