Succinct indexes for strings, binary relations and multilabeled trees

Abstract

We define and design succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. The main advantage of succinct indexes as opposed to succinct (integrated data/index) encodings is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this article, we present succinct indexes for various data types, namely strings, binary relations and multilabeled trees. Given the support for the interface of the ADTs of these data types, we can support various useful operations efficiently by constructing succinct indexes for them. When the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data.Using our techniques, we design a succinct encoding that represents a string of length n over an alphabet of size σ using nH k (S)+lg σ ·o(n)+O(nlg σ /lg lg lg σ ) bits to support access/rank/select operations in o((lg lg σ ) 1+ ) time, for any fixed constant > 0. We also design a succinct text index using nH 0 (S) + O(n lg σ /lg lg σ ) bits that supports finding all the occ occurrences of a given pattern of length m in O(mlg lg σ + occ lg n/ lg σ ) time, for any fixed constant 0 < < 1. Previous results on these two problems either have a lg σ factor instead of lg lg σ in the running time, or are not compressed. Finally, we present succinct encodings of binary relations and multi-labeled trees that are more compact than previous structures.

Stronger Lempel-Ziv Based Compressed Text Indexing

“…Otherwise the time is O( log σ log log u ) and the space is uH 0 (S) + o(u log σ) bits. The representation of Golynski et al [16] requires n(log σ + o(log σ)) = O(n log σ) bits of space [5], allowing us to compute select c in O(1) time, and rank c and access to…”

“…In the RevTrie of Fig. 5(b), it holds that L[i] = 'a' for every i in the preorder interval [2,5]. In the same example, note that if we follow the suffix link ϕ(i), for 2 i 5, we discard the symbol 'a'.…”

Design of Practical Succinct Data Structures for Large Data Collections

“…Systematic SDS have a clear separation of the index from the compressed data, with several advantages [3]. When this separation is not obtained, the resulting SDS are called non-systematic because many bits contribute simultaneously both to the index and the compressed data.…”

Compact Rich-Functional Binary Relation Representations

“…The first representation specifically designed for binary relations [3] supports rank, select and access on the rows (labels) of the relation, for the purpose of supporting faster joins on labels, via a reduction to the rank and select operators on strings, later extended to index text [13], and to separate the content from the index [4], which in turn allows supporting labeled operations on planar and quasi-planar labeled graphs [2].…”