You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
Abstract We consider document listing on string collections, that is, finding in which strings a given pattern appears. In particular, we focus on repetitive collections: a collection of size N over alphabet [ 1 , σ ] is composed of D copies of a string of size n, and s edits are applied on ranges of copies. We introduce the first document listing index with size O ˜ ( n + s ) , precisely O ( ( n lg ⁡ σ + s lg 2 ⁡ N ) lg ⁡ D ) bits, and with useful worst-case time guarantees: Given a pattern of length m, the index reports the ndoc > 0 strings where it appears in time O ( m lg 1 + ϵ ⁡ N ⋅ ndoc ) , for any constant ϵ > 0 (and tells in time O ( m lg ⁡ N ) if ndoc = 0 ). Our technique is to augment a range data structure that is commonly used on grammar-based indexes, so that instead of retrieving all the pattern occurrences, it computes useful summaries on them. We show that the idea has independent interest: we introduce the first grammar-based index that, on a text T [ 1 , N ] with a grammar of size r, uses O ( r lg ⁡ N ) bits and counts the number of occurrences of a pattern P [ 1 , m ] in time O ( m 2 + m lg 2 + ϵ ⁡ r ) , for any constant ϵ > 0 . We also give the first index using O ( z lg ⁡ ( N z ) lg ⁡ N ) bits, where T is parsed by Lempel–Ziv into z phrases, counting occurrences in time O ( m lg 2 + ϵ ⁡ N ) .

Cited Abstract(s):
[1]: We present a novel implementation of compressed suffix arrays exhibiting new tradeoffs between search time and space occupancy for a given text (or sequence) of n symbols over an alphabet σ, where each symbol is encoded by lgvσv bits. We show that compressed suffix arrays use just nH h + σ bits, while retaining full text indexing functionalities, such as searching any pattern sequence of length m in O(m lg vσv + polylog(n)) time. The term H h ≤ lg vσv denotes the hth-order empirical entropy of the text, which means that our index is nearly optimal in space apart from lower-order terms, achieving asymptotically the empirical entropy of the text (with a multiplicative constant 1). If the text is highly compressible so that H n = o(1) and the alphabet size is small, we obtain a text index with o(m) search time that requires only o(n) bits. Further results and tradeoffs are reported in the paper.

[2]: We propose succinct data structures for text retrieval systems supporting document listing queries and ranking queries based on the tf*idf (term frequency times inverse document frequency) scores of documents. Traditional data structures for these problems support queries only for some predetermined keywords. Recently Muthukrishnan proposed a data structure for document listing queries for arbitrary patterns at the cost of data structure size. For computing the tf*idf scores there has been no efficient data structures for arbitrary patterns. Our new data structures support these queries using small space. The space is only 2 @e times the size of compressed documents plus 10n bits for a document collection of length n, for any 0<@e=<1. This is much smaller than the previous O(nlogn) bit data structures. Query time is O(m+qlog^@en) for listing and computing tf*idf scores for all q documents containing a given pattern of length m. Our data structures are flexible in a sense that they support queries for arbitrary patterns.

[3]: Given a set @math of @math strings of total length @math , our task is to report the "most relevant"strings for a given query pattern @math . This involves somewhat more advanced query functionality than the usual pattern matching, as some notion of "most relevant" is involved. In information retrieval literature, this task is best achieved by using inverted indexes. However, inverted indexes work only for some predefined set of patterns. In the pattern matching community, the most popular pattern-matching data structures are suffix trees and suffix arrays. However, a typical suffix tree search involves going through all the occurrences of the pattern over the entire string collection, which might be a lot more than the required relevant documents. The first formal framework to study such kind of retrieval problems was given by [Muthukrishnan, 2002]. He considered two metrics for relevance: frequency and proximity. He took a threshold-based approach on these metrics and gave data structures taking @math words of space. We study this problem in a slightly different framework of reporting the top @math most relevant documents (in sorted order) under similar and more general relevance metrics. Our framework gives linear space data structure with optimal query times for arbitrary score functions. As a corollary, it improves the space utilization for the problems in [Muthukrishnan, 2002] while maintaining optimal query performance. We also develop compressed variants of these data structures for several specific relevance metrics.

[4]: We introduce LZ-End, a new member of the Lempel-Ziv family of text compressors, which achieves compression ratios close to those of LZ77 but is much faster at extracting arbitrary text substrings. We then build the first self-index based on LZ77 (or LZ-End) compression, which in addition to text extraction offers fast indexed searches on the compressed text. This self-index is particularly effective for representing highly repetitive sequence collections, which arise for example when storing versioned documents, software repositories, periodic publications, and biological sequence databases.

[5]: We are given a collection D of text documents d 1 ,…,d k , with ∑ i = n, which may be preprocessed. In the document listing problem, we are given an online query comprising of a pattern string p of length m and our goal is to return the set of all documents that contain one or more copies of p. In the closely related occurrence listing problem, we output the set of all positions within the documents where pattern p occurs. In 1973, Weiner [24] presented an algorithm with O(n) time and space preprocessing following which the occurrence listing problem can be solved in time O(m + output) where output is the number of positions where p occurs; this algorithm is clearly optimal. In contrast, no optimal algorithm is known for the closely related document listing problem, which is perhaps more natural and certainly well-motivated.We provide the first known optimal algorithm for the document listing problem. More generally, we initiate the study of pattern matching problems that require retrieving documents matched by the patterns; this contrasts with pattern matching problems that have been studied more frequently, namely, those that involve retrieving all occurrences of patterns. We consider document retrieval problems that are motivated by online query processing in databases, Information Retrieval systems and Computational Biology. We present very efficient (optimal) algorithms for our document retrieval problems. Our approach for solving such problems involve performing "local" encodings whereby they are reduced to range query problems on geometric objects --- points and lines --- that have color. We present improved algorithms for these colored range query problems that arise in our reductions using the structural properties of strings. This approach is quite general and yields simple, efficient, implementable algorithms for all the document retrieval problems in this paper.
The first optimal-time and linear-space solution to document listing is due to Muthukrishnan [5] , who solves the problem in @math time using an index of @math bits of space. Later solutions [2] [3] improved the space to essentially the statistical entropy of @math , at the price of multiplying the times by low-order polylogs of @math (e.g., @math time with @math bits on top of the entropy [2] [1] ). However, statistical entropy does not capture repetitiveness well [4] , and thus these solutions are not satisfactory in repetitive collections.