Crossing the logarithmic barrier for dynamic boolean data structure lower bounds

This paper proves the first superlogarithmic lower bounds on the cell probe complexity of dynamic Boolean (also known as decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new method for proving dynamic cell probe lower bounds and use it to prove an Ω (lg1.5 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 F2 [M. Patrascu, Lower bounds for 2-dimensional range counting, in STOC 2007, ACM, New York, 2007, pp. 40-46]. Proving an ω (lg n) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Patracscu's obituary [M. Thorup, Bull. Eur. Assoc. Theor. Comput. Sci., 109 (2013), pp. 7-13]. This result also implies the first ω (lg n) lower bound for the classical 2-dimensional (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 (Chebyshev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the "cell sampling"method of Panigrahy, Talwar, and Wieder [Lower bounds on near neighbor search via metric expansion, FOCS 2010, IEEE Computer Society, Los Alamitos, CA, 2010, pp. 805-814].