来自四方通信的摊销动态小区探测下限
Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication
摘要：
本文提出了一种新的技术，证明摊销，随机细胞探针下界的动态数据结构问题。我们引入了一个新的随机非确定性四方通信模型，使“加速”，保留错误的动态数据结构的模拟。我们使用这种技术来证明一个$Ω(n(\log n/\log\log n)^2)$细胞探测下界的动态二维加权正交范围计数问题（2D-ORC）与$n/\mathrm{poly}\log n$更新和$n$查询，即使是数据结构与$\exp(-\tildeΩ(n))$成功概率。该结果不仅证明了迄今为止的最高摊销下限，而且在最强的可能意义上也是紧密的，因为可以通过具有最坏情况操作时间的确定性数据结构获得匹配的上限。这是动态数据结构的“锐阈值”现象的第一次演示。我们更广泛的动机是，细胞探测下限指数小的成功促进减少从动态到静态数据结构。作为一个概念验证，我们表明，我们的下限稍微加强的版本将意味着一个$Ω((\log n /\log\log n)^2)$下限的静态3D-ORC问题与$O(n\log^{O(1)}n)$空间。这样的结果将给出超过最高已知静态单元探测下限的接近二次的改进，并且打破静态数据结构的长期存在的$Ω(\log n)$障碍。
英文摘要：
This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic data structure problems. We introduce a new randomized nondeterministic four-party communication model that enables "accelerated", error-preserving simulations of dynamic data structures. We use this technique to prove an $Ω(n(\log n/\log\log n)^2)$ cell-probe lower bound for the dynamic 2D weighted orthogonal range counting problem (2D-ORC) with $n/\mathrm{poly}\log n$ updates and $n$ queries, that holds even for data structures with $\exp(-\tildeΩ(n))$ success probability. This result not only proves the highest amortized lower bound to date, but is also tight in the strongest possible sense, as a matching upper bound can be obtained by a deterministic data structure with worst-case operational time. This is the first demonstration of a "sharp threshold" phenomenon for dynamic data structures. Our broader motivation is that cell-probe lower bounds for exponentially small success facilitate reductions from dynamic to static data structures. As a proof-of-concept, we show that a slightly strengthened version of our lower bound would imply an $Ω((\log n /\log\log n)^2)$ lower bound for the static 3D-ORC problem with $O(n\log^{O(1)}n)$ space. Such result would give a near quadratic improvement over the highest known static cell-probe lower bound, and break the long standing $Ω(\log n)$ barrier for static data structures.
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