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We consider the {\it indexable dictionary} problem, which consists of storinga set $S \subseteq \{0,...,m-1\}$ for some integer $m$, while supporting theoperations of $\Rank(x)$, which returns the number of elements in $S$ that areless than $x$ if $x \in S$, and -1 otherwise; and $\Select(i)$ which returnsthe $i$-th smallest element in $S$. We give a data structure that supports bothoperations 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 $n$-elementsubset from a universe of size $m$. Previous dictionaries taking this spaceonly supported (yes/no) membership queries in O(1) time. In the cell probemodel 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 datastructure, including:  An information-theoretically optimal representation of a $k$-ary cardinaltree that supports standard operations in constant time,  A representation of a multiset of size $n$ from $\{0,...,m-1\}$ in ${\calB}(n,m+n) + o(n)$ 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 constanttime.

Succinct Indexable Dictionaries with Applications to Encoding $k$-ary Trees, Prefix Sums and Multisets