Abstract. The proliferation of online text, such as found on the World Wide Web and in online databases, motivates the need for space-efficient text indexing methods that support fast string searching. We model this scenario as follows: Consider a text T consisting of n symbols drawn from a fixed alphabet Σ. The text T can be represented in n lg |Σ | bits by encoding each symbol with lg |Σ | bits. The goal is to support fast online queries for searching any string pattern P of m symbols, with T being fully scanned only once, namely, when the index is created at preprocessing time. The text indexing schemes published in the literature are greedy in terms of space usage: they require Ω(n lg n) additional bits of space in the worst case. For example, in the standard unit cost RAM, suffix trees and suffix arrays need Ω(n) memory words, each of Ω(lg n) bits. These indexes are larger than the text itself by a multiplicative factor of Ω(lg |Σ | n), which is significant when Σ is of constant size, such as in ascii or unicode. On the other hand, these indexes support fast searching, either in O(m lg |Σ|) timeorinO(m +lgn) time, plus an output-sensitive cost O(occ) for listing the occ pattern occurrences. We present a new text index that is based upon compressed representations of suffix arrays and suffix trees. It achieves a fast O(m / lg |Σ | n +lgɛ |Σ | n) search time in the worst case, for any constant

Full-text indexes provide fast substring search over large text collections. A serious problem of these indexes has traditionally been their space consumption. A recent trend is to develop indexes that exploit the compressibility of the text, so that their size is a function of the compressed text length. This concept has evolved into self-indexes, which in addition contain enough information to reproduce any text portion, so they replace the text. The exciting possibility of an index that takes space close to that of the compressed text, replaces it, and in addition provides fast search over it, has triggered a wealth of activity and produced surprising results in a very short time, and radically changed the status of this area in less than five years. The most successful indexes nowadays are able to obtain almost optimal space and search time simultaneously. In this paper we present the main concepts underlying self-indexes. We explain the relationship between text entropy and regularities that show up in index structures and permit compressing them. Then we cover the most relevant self-indexes up to date, focusing on the essential aspects on how they exploit the text compressibility and how they solve efficiently various search problems. We aim at giving the theoretical background to understand and follow the developments in this area.

Abstract. Given a sequence S = s1s2... sn of integers smaller than r = O(polylog(n)), we show how S can be represented using nH0(S) + o(n) bits, so that we can know any sq, as well as answer rank and select queries on S, in constant time. H0(S) is the zero-order empirical entropy of S and nH0(S) provides an Information Theoretic lower bound to the bit storage of any sequence S via a fixed encoding of its symbols. This extends previous results on binary sequences, and improves previous results on general sequences where those queries are answered in O(log r) time. For larger r, we can still represent S in nH0(S) + o(n log r) bits and answer queries in O(log r / log log n) time. Another contribution of this paper is to show how to combine our compressed representation of integer sequences with an existing compression boosting technique to design compressed full-text indexes that scale well with the size of the input alphabet Σ. Namely, we design a variant of the FM-index that indexes a string T [1, n] within nHk(T) + o(n) bits of storage, where Hk(T) is the k-th order empirical entropy of T. This space bound holds simultaneously for all k ≤ α log |Σ | n, constant 0 < α < 1, and |Σ | = O(polylog(n)). This index counts the occurrences of an arbitrary pattern P [1, p] as a substring of T in O(p) time; it locates each pattern occurrence in O(log 1+ε n) time, for any constant 0 < ε < 1; and it reports a text substring of length ℓ in O(ℓ + log 1+ε n) time.

A succinct full-text self-index is a data structure built on a text T = t1t2...tn, which takes little space (ideally close to that of the compressed text), permits efficient search for the occurrences of a pattern P = p1p2... pm in T, and is able to reproduce any text substring, so the self-index replaces the text. Several remarkable self-indexes have been developed in recent years. Many of those take space proportional to nH0 or nHk bits, where Hk is the kth order empirical entropy of T. The time to count how many times does P occur in T ranges from O(m) to O(m log n). In this paper we present a new self-index, called RLFM index for “run-length FM-index”, that counts the occurrences of P in T in O(m) time when the alphabet size is σ = O(polylog(n)). The RLFM index requires nHk log σ + O(n) bits of space, for any k ≤ α log σ n and constant 0 < α < 1. Previous indexes that achieve O(m) counting time either require more than nH0 bits of space or require that σ = O(1). We also show that the RLFM index can be enhanced to locate occurrences in the text and display text substrings in time independent of σ. In addition, we prove a close relationship between the kth order entropy of the text and some regularities that show up in their suffix arrays and in the Burrows-Wheeler transform of T. This relationship is of independent interest and permits bounding the space occupancy of the RLFM index, as well as that of other existing compressed indexes. Finally, we present some practical considerations in order to implement the RLFM index, obtaining two implementations with different space-time tradeoffs. We empirically compare our indexes against the best existing implementations and show that they are practical and competitive against those. 1

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Consider an ordered, static tree T on t nodes where each node has a label from alphabet set Σ. TreeTmaybeofar bitrary degree and of arbitrary shape. Say, we wish to support basic navigational operations such as find the parent of node u,theith child of u, and any child of u with label α. In a seminal work over fifteen years ago, Jacobson [15] observed that pointer-based tree representations are wasteful in space and introduced the notion of succinct data structures. He studied the special case of unlabeled trees and presented a succinct data structure of 2t+o(t) bits supporting navigational operations in O(1) time. The space used is asymptotically optimal with the information-theoretic lower bound averaged over all trees. This led to a slew of results on succinct data structures for arrays, trees, strings

Abstract. We show that, by combining an existing compression boosting technique with the wavelet tree data structure, we are able to design a variant of the FM-index which scales well with the size of the input alphabet Σ. The size of the new index built on a string T [1, n] is bounded by nHk(T)+O � (n log log n) / log |Σ | n � bits, where Hk(T) is the k-th order empirical entropy of T. The above bound holds simultaneously for all k ≤ α log |Σ | n and 0 < α < 1. Moreover, the index design does not depend on the parameter k, which plays a role only in analysis of the space occupancy. Using our index, the counting of the occurrences of an arbitrary pattern P [1, p] as a substring of T takes O(p log |Σ|) time. Locating each pattern occurrence takes O(log |Σ | (log 2 n / log log n)) time. Reporting a text substring of length ℓ takes O((ℓ + log 2 n / log log n) log |Σ|) time. 1

We report on a new and improved version of high-order entropy-compressed suffix arrays, which has theoretical performance guarantees similar to those in our earlier work [16], yet represents an improvement in practice. Our experiments indicate that the resulting text index offers state-of-the-art compression. In particular, we require roughly 20 % of the original text size—without requiring a separate instance of the text—and support fast and powerful searches. To our knowledge, this is the best known method in terms of space for fast searching. 1

We define and design succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures called succinct indexes that 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. As opposed to succinct encodings, The main advantage of succinct indexes 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 paper, we present succinct indexes for various data types, namely strings, binary relations and multi-labeled 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 the first succinct encoding that represents a string of length n over alphabet [σ] using nHk + o(n lg σ) bits 1 that support access / rank / select operations in o((lg lg σ) 3) time. We also design the first succinct text index using nHk + o(n lg σ) bits that supports pattern matching queries in O(m lg lg σ + occ lg 1+ɛ n lg lg σ) time, for a given pattern of length m. Previous results on these two problems either have a lg σ factor instead of lg lg σ in terms of running time, or are not compressible, but our results do not have such problems. More results are reported in the paper. 1 We use lg n to denote ⌈log2 n⌉. 1

Research on succinct data structures has made significant progress in recent years. An essential building block of many of those techniques is a data structure to perform rank and select operations over a bit array. The first operation tells how many bits are set up to some position, and the second the position of the i-th bit set. Albeit there exist constanttime solutions that require sublinear extra space, the practicality of those solutions against more naive ones has not been carefully studied. In this paper we show some results in this respect, which suggest that in many practical cases the simpler solutions are better in terms of time and extra space.