论文标题
动态分数级联的下限
A Lower Bound for Dynamic Fractional Cascading
论文作者
论文摘要
我们研究了数据结构中基本思想之一的局限性:分数级联。这是一项重要的数据结构技术,可以加快多个列表中相同键的重复搜索,并且具有许多应用程序。具体而言,输入是一个恒定度的“目录”图,$ g $,以及分配给$ g $的每个顶点的值列表。目的是预处理输入,以便给定连接的子图$ h $的$ g $和单个查询值$ q $,人们可以在每个列表中的$ q $的前身,属于$ \ scat $。 Chazelle和Guibas的经典结果表明,在指针机中,可以在$Ø(\ log N + | \ scat |)$的最佳时间内完成,其中$ n $是值的总数。但是,如果允许插入和删除值,则查询时间将变慢为$Ø(\ log n + | \ scat | \ log \ log \ log n)$。如果仅允许插入(或删除),那么再次,可以在更新时使用摊销,可以获得最佳查询时间。 我们证明了$ω(\ log n \ sqrt {\ log \ log n})$的下限在最差的动态分数级联时,当查询是长度$ o(\ log log n)$的路径时。下边界都适用于具有摊销的Polygarithmic更新时间和增量数据结构的完全动态数据结构,并带有Polyogarithmic糟糕的更新时间。作为一方面,这也言论摊销对于获得最佳的增量数据结构至关重要。 这是为打破$ω(\ log n)$屏障的动态数据结构的第一个非平凡指针机的下限。为了获得此结果,我们开发了许多新的想法和技术,希望可以在指针机模型中获得其他动态下限有用。
We investigate the limits of one of the fundamental ideas in data structures: fractional cascading. This is an important data structure technique to speed up repeated searches for the same key in multiple lists and it has numerous applications. Specifically, the input is a "catalog" graph, $G$, of constant degree together with a list of values assigned to every vertex of $G$. The goal is to preprocess the input such that given a connected subgraph $H$ of $G$ and a single query value $q$, one can find the predecessor of $q$ in every list that belongs to $\scat$. The classical result by Chazelle and Guibas shows that in a pointer machine, this can be done in the optimal time of $Ø(\log n + |\scat|)$ where $n$ is the total number of values. However, if insertion and deletion of values are allowed, then the query time slows down to $Ø(\log n + |\scat| \log\log n)$. If only insertions (or deletions) are allowed, then once again, an optimal query time can be obtained but by using amortization at update time. We prove a lower bound of $Ω( \log n \sqrt{\log\log n})$ on the worst-case query time of dynamic fractional cascading, when queries are paths of length $O(\log n)$. The lower bound applies both to fully dynamic data structures with amortized polylogarithmic update time and incremental data structures with polylogarithmic worst-case update time. As a side, this also roves that amortization is crucial for obtaining an optimal incremental data structure. This is the first non-trivial pointer machine lower bound for a dynamic data structure that breaks the $Ω(\log n)$ barrier. In order to obtain this result, we develop a number of new ideas and techniques that hopefully can be useful to obtain additional dynamic lower bounds in the pointer machine model.