简洁可索引字典及其在$k$ -叉树、前缀和与多集编码中的应用
Succinct Indexable Dictionaries with Applications to Encoding $k$-ary Trees, Prefix Sums and Multisets
摘要：
我们考虑{ \it可索引字典}问题，该问题包括为某个整数$m$存储一个集合$S \subseteq \{0,...,m-1\}$，同时支持$\Rank(x)$的操作，如果$x \in S$，则返回$S$中小于$x$的元素的数量，否则返回-1;以及$\Select(i)$，返回$S$中第$i$个最小元素。我们给出了一个数据结构，支持在O（1）时间的RAM模型上的操作，并要求${\cal B}(n,m) + o(n) + O(\lg \lg m)$位存储一组大小为$n$，其中${\cal B}(n,m) = \ceil{\lg {m \choose n}}$是存储任何$n$元素子集所需的最小位数从一个宇宙的大小$m$。以前占用这个空间的字典只支持（yes/no）成员查询，时间复杂度为O（1）。在细胞探针模型中，我们可以删除空间边界中的$O(\lg \lg m)$加法项，回答了Fich和Miltersen以及Pagh提出的问题。我们提出了我们的索引字典数据结构的扩展和应用，包括：一种支持常数时间标准操作的$k$元基数树的信息论最优表示，一种以${\cal B}(n,m+n) + o(n)$位表示的来自$\{0,...,m-1\}$的大小为$n$的多集表示，支持在恒定时间内的$\Rank$和$\Select$操作（的适当推广），以及$n$非负整数序列的表示，其以${\cal B}(n,m+n) + o(n)$位求和为$m$，其支持在恒定时间内的前缀和查询。
英文摘要：
We consider the {\it indexable dictionary} problem, which consists of storing a set $S \subseteq \{0,...,m-1\}$ for some integer $m$, while supporting the operations of $\Rank(x)$, which returns the number of elements in $S$ that are less than $x$ if $x \in S$, and -1 otherwise; and $\Select(i)$ which returns the $i$-th smallest element in $S$. We give a data structure that supports both operations in O(1) time on the RAM model and requires ${\cal B}(n,m) + o(n) + O(\lg \lg m)$ bits to store a set of size $n$, where ${\cal B}(n,m) = \ceil{\lg {m \choose n}}$ is the minimum number of bits required to store any $n$-element subset from a universe of size $m$. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the $O(\lg \lg m)$ additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: An information-theoretically optimal representation of a $k$-ary cardinal tree that supports standard operations in constant time, A representation of a multiset of size $n$ from $\{0,...,m-1\}$ in ${\cal B}(n,m+n) + o(n)$ bits that supports (appropriate generalizations of) $\Rank$ and $\Select$ operations in constant time, and A representation of a sequence of $n$ non-negative integers summing up to $m$ in ${\cal B}(n,m+n) + o(n)$ bits that supports prefix sum queries in constant time.
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