An Adaptive Step Toward the Multiphase Conjecture

In 2010, P\v{a}tra\c{s}cu proposed the following three-phase dynamic problem, as a candidate for proving polynomial lower bounds on the operational time of dynamic data structures: I: Preprocess a collection of sets $\vec{S} = S1, \ldots , Sk \subseteq [n]$, where $k=\operatorname{poly}(n)$. II: A set $T\subseteq [n]$ is revealed, and the data structure updates its memory. III: An index $i \in [k]$ is revealed, and the data structure must determine if $S_i\cap T=^? \emptyset$. P\v{a}tra\c{s}cu conjectured that any data structure for the Multiphase problem must make $n^\epsilon$ cell-probes in either Phase II or III, and showed that this would imply similar unconditional lower bounds on many important dynamic data structure problems. Alas, there has been almost no progress on this conjecture in the past decade since its introduction. We show an $\tilde{\Omega}(\sqrt{n})$ cell-probe lower bound on the Multiphase problem for data structures with general (adaptive) updates, and queries with unbounded but "layered" adaptivity. This result captures all known set-intersection data structures and significantly strengthens previous Multiphase lower bounds, which only captured non-adaptive data structures. Our main technical result is a communication lower bound on a 4-party variant of P\v{a}tra\c{s}cu's Number-On-Forehead Multiphase game, using information complexity techniques. We also show that a lower bound on P\v{a}tra\c{s}cu's original NOF game would imply a polynomial ($n^{{1+\epsilon}$)} lower bound on the number of wires of any constant-depth circuit with arbitrary gates computing a random $\tilde{O}(n)\times n$ linear operator $x \mapsto Ax$, a long-standing open problem in circuit complexity. This suggests that the NOF conjecture is much stronger than its data structure counterpart.