TY - GEN

T1 - Crossing the logarithmic barrier for dynamic boolean data structure lower bounds

AU - Larsen, Kasper Green

AU - Weinstein, Omri

AU - Yu, Huacheng

N1 - Publisher Copyright:
© 2018 Association for Computing Machinery.

PY - 2018/6/20

Y1 - 2018/6/20

N2 - 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 approach and use it to prove a 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 (Pǎtraşcu, 2007). Proving an (lg n) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu’s obituary (Thorup, 2013). This result also implies the first (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 (Chebyshev) polynomials which May be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling" method of Panigrahy et al. (2010).

AB - 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 approach and use it to prove a 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 (Pǎtraşcu, 2007). Proving an (lg n) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu’s obituary (Thorup, 2013). This result also implies the first (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 (Chebyshev) polynomials which May be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling" method of Panigrahy et al. (2010).

KW - Cell probe complexity

KW - Data structures

KW - Dynamic problems

KW - Lower bounds

KW - Range searching

UR - http://www.scopus.com/inward/record.url?scp=85049904907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049904907&partnerID=8YFLogxK

U2 - 10.1145/3188745.3188790

DO - 10.1145/3188745.3188790

M3 - Conference contribution

AN - SCOPUS:85049904907

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 485

EP - 492

BT - STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

A2 - Henzinger, Monika

A2 - Kempe, David

A2 - Diakonikolas, Ilias

PB - Association for Computing Machinery

T2 - 50th Annual ACM Symposium on Theory of Computing, STOC 2018

Y2 - 25 June 2018 through 29 June 2018

ER -