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]: Many document collections consist largely of repeated material, and several indexes have been designed to take advantage of this. There has been only preliminary work, however, on document retrieval for repetitive collections. In this paper we show how one of those indexes, the run-length compressed suffix array (RLCSA), can be extended to support document listing. In our experiments, our additional structures on top of the RLCSA can reduce the query time for document listing by an order of magnitude while still using total space that is only a fraction of the raw collection size. As a byproduct, we develop a new document listing technique for general collections that is of independent interest.
The last previous work is PDL [1] , which stores inverted lists at sampled nodes in the suffix tree of @math , and then grammar-compresses the set of inverted lists. For a sampling step @math , it requires @math bits plus the (unbounded) space of the inverted lists. Searches that lead to the sampled nodes have their answers precomputed, whereas the others cover a suffix array range of size @math and are solved by brute force in time @math .