基于讹误的半自适应数据结构的下界
Lower Bounds for Semi-adaptive Data Structures via Corruption
摘要：
在动态数据结构问题中，我们希望在内存中保持一些数据的编码，以这种方式，我们可以有效地执行一系列查询和更新数据。一个长期存在的开放问题在这一领域是证明一个无条件多项式下界的更新时间和查询时间的自适应动态数据结构计算一些显式函数之间的权衡。Ko和Weinstein为计算不相交函数的{ \em半自适应\}数据结构的受限类提供了这样的下限。在那里，数据是$\{1,\dots,n\}$的子集$x_1,\dots,x_k$和$y$，更新可以修改$y$（通过插入和移除元素），并且查询是索引$i \in \{1,\dots,k\}$（查询$i$应该回答$x_i$和$y$是否不相交，即，它应该计算应用于$(x_i, y)$的不相交度函数）。半自适应性限制了如何访问数据结构以回答查询。我们推广了Ko和Weinstein的下界，不仅适用于不相交性，而且适用于任何在光滑腐败界下具有高复杂性的函数。
英文摘要：
In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove an unconditional polynomial lower bound of a trade-off between the update time and the query time of an adaptive dynamic data structure computing some explicit function. Ko and Weinstein provided such lower bound for a restricted class of {\em semi-adaptive\} data structures, which compute the Disjointness function. There, the data are subsets $x_1,\dots,x_k$ and $y$ of $\{1,\dots,n\}$, the updates can modify $y$ (by inserting and removing elements), and the queries are an index $i \in \{1,\dots,k\}$ (query $i$ should answer whether $x_i$ and $y$ are disjoint, i.e., it should compute the Disjointness function applied to $(x_i, y)$). The semi-adaptiveness places a restriction in how the data structure can be accessed in order to answer a query. We generalize the lower bound of Ko and Weinstein to work not just for the Disjointness, but for any function having high complexity under the smooth corruption bound.
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