Omri Weinstein, Columbia University

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, a longstanding open question in circuit complexity.

Joint work with Young Kun Ko.