Adapt or Die：非自适应动态数据结构的多项式下界
Adapt or Die: Polynomial Lower Bounds for Non-Adaptive Dynamic Data Structures
摘要：
在本文中，我们研究的作用，非自适应维护动态数据结构。粗略地说，如果数据结构在处理查询或更新时读取和/或写入的存储器位置仅取决于查询或更新而不取决于先前读取的单元的内容，则该数据结构是非自适应的。我们研究这种非自适应数据结构的细胞探针模型。该模型是限制最少的下限模型之一，并且特别地，单元探测下限适用于在流行的字-RAM模型中开发的数据结构。不幸的是，这种普遍性的代价很高：任何数据结构问题的最高下界都只是多对数的。我们的主要结果是证明，事实上可以获得多项式细胞探测下界的非自适应数据结构。为了更清楚地揭示看似固有的多对数下限障碍，我们研究了几种不同的概念，非自适应性和确定的关键属性，必须处理，如果我们要证明多项式下限没有限制的数据结构。最后，我们的结果还揭示了数据结构和深度2电路之间的有趣联系。这使我们能够翻译成好的候选人高电路下限的复杂的硬数据结构问题，特别是在该地区的线性电路的线性算子。在限制性稍强的模型中建立数据结构的下界证明，我们还提出了一些我们认为值得在电路下界领域进行研究的线性算子的性质。
英文摘要：
In this paper, we study the role non-adaptivity plays in maintaining dynamic data structures. Roughly speaking, a data structure is non-adaptive if the memory locations it reads and/or writes when processing a query or update depend only on the query or update and not on the contents of previously read cells. We study such non-adaptive data structures in the cell probe model. This model is one of the least restrictive lower bound models and in particular, cell probe lower bounds apply to data structures developed in the popular word-RAM model. Unfortunately, this generality comes at a high cost: the highest lower bound proved for any data structure problem is only polylogarithmic. Our main result is to demonstrate that one can in fact obtain polynomial cell probe lower bounds for non-adaptive data structures. To shed more light on the seemingly inherent polylogarithmic lower bound barrier, we study several different notions of non-adaptivity and identify key properties that must be dealt with if we are to prove polynomial lower bounds without restrictions on the data structures. Finally, our results also unveil an interesting connection between data structures and depth-2 circuits. This allows us to translate conjectured hard data structure problems into good candidates for high circuit lower bounds; in particular, in the area of linear circuits for linear operators. Building on lower bound proofs for data structures in slightly more restrictive models, we also present a number of properties of linear operators which we believe are worth investigating in the realm of circuit lower bounds.
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