Abstract: In 2010, Patrascu proposed the Multiphase problem, as a candidate for proving polynomial lower bounds on the operational time of dynamic data structures. Patrascu conjectured that any data structure for the Multiphase problem must make n^eps cell-probes in either the update or query phase, and showed that this would imply similar unconditional lower bounds on many important dynamic data structure problems. There has been almost no progress on this conjecture in the past decade. We show an ~\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 Patrascu's NOF Multiphase game, using information complexity techniques. We show that this communication lower bound implies the first polynomial (n^{1+1/k}) lower bound on the number of wires required to compute a *linear* operator using *nonlinear* (degree-k) gates. This gives a partial answer to a longstanding open question in circuit complexity.

Joint work with Young Kun Ko. 

Bio: Omri Weinstein is an assistant professor in the theoretical computer science group at Columbia University. He is mainly interested in the interplay between information theory, complexity and data structures. He was a graduate student at Princeton University (2015), where he was lucky to have Mark Braverman as my advisor. Before joining Columbia, he was a Simons Society Junior Fellow at Courant Institute (NYU).