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Seminar: Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds

Abstract: 

We prove the first super-logarithmic lower bounds on the cell-probe complexity
of dynamic *boolean* (a.k.a. decision) data structure problems, a long-standing
milestone in data structure lower bounds.

We introduce a new technique and use it to prove a (log^{1.5} n) lower bound on the
operational time of a wide range of boolean data structure problems, most notably,
on the query time of dynamic range counting *over F_2* ([Patrascu07]). Proving an
\omega(log n) lower bound for this problem was explicitly posed as one of five important
open problems in the late Mihai Patrascu's obituary [Thorup13]. This also implies the
first \omega(log n) lower bound for the classical 2D range counting problem, one of the
most fundamental data structure problems in computational geometry and spatial databases.
We derive similar lower bounds for boolean versions of dynamic "polynomial evaluation"
and "2D rectangle stabbing", and for the (non-boolean) problems of "range selection"
and "range median".

Our technical centerpiece is a new way of ``weakly" simulating dynamic data structures
using efficient *one-way* communication protocols with small advantage over random
guessing. This simulation involves a surprising excursion to low-degree (Chebychev)
polynomials which may be of independent interest, and offers an entirely new algorithmic
angle on the "cell sampling" method of Panigrahy et al. [PTW10].

Joint work with Kasper Green Larsen and Huacheng Yu.

Speaker: 
Omri Weinstein
Location: 
CooRE 431
Event Date: 
11/08/2017 - 11:00am
Committee: 
Rutgers/DIMACS Theory of Computing
Event Type: 
Seminar
Organization: 
Columbia University