An Adaptive Step Toward the Multiphase Conjecture

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 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. Alas, there has been almost no progress on this conjecture in the past decade since its introduction. 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 Number-On-Forehead Multiphase game, using information complexity techniques.

Joint work with Young Kun Ko.