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Computer Science > Data Structures and Algorithms
[Submitted on 10 Mar 2017]
Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds
Kasper Green Larsen, Omri Weinstein, Huacheng Yu
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 $tilde{Omega}(log^{1.5} n)$ 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 $mathbb{F}_2$ ([Pat07]). Proving an $omega(lg n)$ 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 $omega(lg n)$ 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].
Subjects:
Data Structures and Algorithms (cs.DS); Computational Complexity (cs.CC); Computational Geometry (cs.CG); Information Theory (cs.IT)
Cite as:
arXiv:1703.03575 [cs.DS]
(or arXiv:1703.03575v1 [cs.DS] for this version)
Submission history
From: Kasper Green Larsen [view email]
[v1] Fri, 10 Mar 2017 08:35:24 UTC (126 KB)
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Kasper Green Larsen
Omri Weinstein
Huacheng Yu
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