This paper proves 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 method for proving dynamic cell probe lower bounds and use it to prove a Ω(lg1.5n) 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 ([Pat07]). Proving an ω(lg n) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu's obituary [Tho13]. This result also implies the first ω(lg 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].
|Original language||American English|
|Title of host publication||2018 Information Theory and Applications Workshop, ITA 2018|
|Publisher||Institute of Electrical and Electronics Engineers Inc.|
|State||Published - 23 Oct 2018|
|Event||2018 Information Theory and Applications Workshop, ITA 2018 - San Diego, United States|
Duration: 11 Feb 2018 → 16 Feb 2018
|Name||2018 Information Theory and Applications Workshop, ITA 2018|
|Conference||2018 Information Theory and Applications Workshop, ITA 2018|
|Period||11/02/18 → 16/02/18|
Bibliographical noteFunding Information:
∗Department of Computer Science, Aarhus University. Supported by MADALGO, grant DNRF84, a Villum Young Investigator Grant and an AUFF Starting Grant. †Department of Computer Science, Columbia University. ‡Department of Computer Science, Stanford University. Supported by NSF CCF-1212372.
© 2018 IEEE.