Succinct Indexable Dictionaries with Applications to Encoding $k$-ary Trees, Prefix Sums and Multisets (0705.0552v1)

Abstract: 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.

• The paper presents a space-efficient dictionary that supports constant-time rank and select operations while using space close to the information-theoretic minimum.
• It employs innovative techniques like MSB bucketing and quotienting to partition the data and reduce the universe size for efficient querying.
• The methods enable practical encoding of k-ary trees, prefix sums, and multisets, advancing the development of succinct data structures in various applications.

An Analysis of Succinct Indexable Dictionaries

The paper "Succinct Indexable Dictionaries with Applications to Encoding $k$-ary Trees, Prefix Sums, and Multisets" by Rajeev Raman, Venkatesh Raman, and S. Srinivasa Rao introduces a significant advancement in the field of succinct data structures, specifically focusing on the indexable dictionary problem. This work addresses the challenge of compactly representing a set of $n$ elements from a universe of size $m$ while supporting efficient retrieval operations.

Overview of Contributions

The authors present a data structure that not only stores the set $S \subseteq \{0, \ldots, m-1\}$ using space close to the information-theoretic minimum but also supports constant-time operations for rank queries and retrieval of the $i$-th smallest element. Specifically, the data structure requires ${\cal B}(n,m) + o(n) + O(\lg \lg m)$ bits, where ${\cal B}(n,m) = \lceil \lg {m \choose n} \rceil$ is the minimum number of bits necessary to uniquely represent any $n$-element subset of a universe of size $m$.

Key Techniques and Concepts

1. MSB Bucketing: A pivotal component of the proposed solution is the most-significant-bit first (MSB) bucketing technique. It enables partitioning elements based on their most significant bits, allowing for significant space savings without sacrificing query efficiency.
2. Quotienting and Universe Reduction: The work leverages advanced hashing techniques, including quotienting and distinguishers, to map elements to a reduced universe efficiently, which is critical for achieving the desired space bounds.
3. Succinct Representation of Multi-dictionaries: The paper extends its techniques to efficiently represent multiple dictionaries, which can have applications in encoding data structures like $k$-ary trees and multisets, crucially supporting operations such as parent-child navigation and rank queries.
4. Fully Indexable Dictionaries (FIDs): The authors extend the utility of FIDs, which support rank and select operations over both the set and its complement, providing a versatile tool for addressing a variety of data representation challenges.

Applications and Implications

The methods developed in this paper are not only theoretically sound but also have wide-ranging applications. For example, representing $k$-ary trees using these techniques allows for constant-time navigational queries in a space-efficient manner. This has implications for compact data representation in fields such as text indexing, computational biology, and network routing, where the underlying structures can be modeled as trees or graphs.

Future Directions

The paper opens several avenues for further exploration. One potential area is the dynamization of the indexable dictionaries, as the current scope is limited to static structures. Furthermore, the pursuit of reducing the lower-order terms in the RAM model could lead to even more space-efficient representations, bringing them closer to the cell probe model’s theoretical limits.

Conclusion

In conclusion, the paper "Succinct Indexable Dictionaries with Applications to Encoding $k$-ary Trees, Prefix Sums, and Multisets" provides a robust framework for compactly storing and querying large datasets. By enhancing our understanding and implementation of succinct data structures, it lays the groundwork for future advancements in efficient data management across numerous applications. This work represents a cornerstone in the ongoing development of space-efficient data structures, promising impactful advancements in both theoretical and applied computer science.