3,462 research outputs found
Combined Data Structure for Previous- and Next-Smaller-Values
Let be a static array storing elements from a totally ordered set. We
present a data structure of optimal size at most
bits that allows us to answer the following queries on in constant time,
without accessing : (1) previous smaller value queries, where given an index
, we wish to find the first index to the left of where is strictly
smaller than at , and (2) next smaller value queries, which search to the
right of . As an additional bonus, our data structure also allows to answer
a third kind of query: given indices , find the position of the minimum in
. Our data structure has direct consequences for the space-efficient
storage of suffix trees.Comment: to appear in Theoretical Computer Scienc
String Indexing with Compressed Patterns
Given a string S of length n, the classic string indexing problem is to preprocess S into a compact data structure that supports efficient subsequent pattern queries. In this paper we consider the basic variant where the pattern is given in compressed form and the goal is to achieve query time that is fast in terms of the compressed size of the pattern. This captures the common client-server scenario, where a client submits a query and communicates it in compressed form to a server. Instead of the server decompressing the query before processing it, we consider how to efficiently process the compressed query directly. Our main result is a novel linear space data structure that achieves near-optimal query time for patterns compressed with the classic Lempel-Ziv 1977 (LZ77) compression scheme. Along the way we develop several data structural techniques of independent interest, including a novel data structure that compactly encodes all LZ77 compressed suffixes of a string in linear space and a general decomposition of tries that reduces the search time from logarithmic in the size of the trie to logarithmic in the length of the pattern
Optimal-Time Text Indexing in BWT-runs Bounded Space
Indexing highly repetitive texts --- such as genomic databases, software
repositories and versioned text collections --- has become an important problem
since the turn of the millennium. A relevant compressibility measure for
repetitive texts is , the number of runs in their Burrows-Wheeler Transform
(BWT). One of the earliest indexes for repetitive collections, the Run-Length
FM-index, used space and was able to efficiently count the number of
occurrences of a pattern of length in the text (in loglogarithmic time per
pattern symbol, with current techniques). However, it was unable to locate the
positions of those occurrences efficiently within a space bounded in terms of
. Since then, a number of other indexes with space bounded by other measures
of repetitiveness --- the number of phrases in the Lempel-Ziv parse, the size
of the smallest grammar generating the text, the size of the smallest automaton
recognizing the text factors --- have been proposed for efficiently locating,
but not directly counting, the occurrences of a pattern. In this paper we close
this long-standing problem, showing how to extend the Run-Length FM-index so
that it can locate the occurrences efficiently within space (in
loglogarithmic time each), and reaching optimal time within
space, on a RAM machine of bits. Within
space, our index can also count in optimal time .
Raising the space to , we support count and locate in
and time, which is optimal in the
packed setting and had not been obtained before in compressed space. We also
describe a structure using space that replaces the text and
extracts any text substring of length in almost-optimal time
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String Indexing for Patterns with Wildcards
We consider the problem of indexing a string of length to report the
occurrences of a query pattern containing characters and wildcards.
Let be the number of occurrences of in , and the size of
the alphabet. We obtain the following results.
- A linear space index with query time .
This significantly improves the previously best known linear space index by Lam
et al. [ISAAC 2007], which requires query time in the worst case.
- An index with query time using space , where is the maximum number of wildcards allowed in the pattern.
This is the first non-trivial bound with this query time.
- A time-space trade-off, generalizing the index by Cole et al. [STOC 2004].
We also show that these indexes can be generalized to allow variable length
gaps in the pattern. Our results are obtained using a novel combination of
well-known and new techniques, which could be of independent interest
Efficient LZ78 factorization of grammar compressed text
We present an efficient algorithm for computing the LZ78 factorization of a
text, where the text is represented as a straight line program (SLP), which is
a context free grammar in the Chomsky normal form that generates a single
string. Given an SLP of size representing a text of length , our
algorithm computes the LZ78 factorization of in time
and space, where is the number of resulting LZ78 factors.
We also show how to improve the algorithm so that the term in the
time and space complexities becomes either , where is the length of the
longest LZ78 factor, or where is a quantity
which depends on the amount of redundancy that the SLP captures with respect to
substrings of of a certain length. Since where
is the alphabet size, the latter is asymptotically at least as fast as
a linear time algorithm which runs on the uncompressed string when is
constant, and can be more efficient when the text is compressible, i.e. when
and are small.Comment: SPIRE 201
Fast Label Extraction in the CDAWG
The compact directed acyclic word graph (CDAWG) of a string of length
takes space proportional just to the number of right extensions of the
maximal repeats of , and it is thus an appealing index for highly repetitive
datasets, like collections of genomes from similar species, in which grows
significantly more slowly than . We reduce from to
the time needed to count the number of occurrences of a pattern of
length , using an existing data structure that takes an amount of space
proportional to the size of the CDAWG. This implies a reduction from
to in the time needed to
locate all the occurrences of the pattern. We also reduce from
to the time needed to read the characters of the
label of an edge of the suffix tree of , and we reduce from
to the time needed to compute the matching
statistics between a query of length and , using an existing
representation of the suffix tree based on the CDAWG. All such improvements
derive from extracting the label of a vertex or of an arc of the CDAWG using a
straight-line program induced by the reversed CDAWG.Comment: 16 pages, 1 figure. In proceedings of the 24th International
Symposium on String Processing and Information Retrieval (SPIRE 2017). arXiv
admin note: text overlap with arXiv:1705.0864
The Rightmost Equal-Cost Position Problem
LZ77-based compression schemes compress the input text by replacing factors
in the text with an encoded reference to a previous occurrence formed by the
couple (length, offset). For a given factor, the smallest is the offset, the
smallest is the resulting compression ratio. This is optimally achieved by
using the rightmost occurrence of a factor in the previous text. Given a cost
function, for instance the minimum number of bits used to represent an integer,
we define the Rightmost Equal-Cost Position (REP) problem as the problem of
finding one of the occurrences of a factor which cost is equal to the cost of
the rightmost one. We present the Multi-Layer Suffix Tree data structure that,
for a text of length n, at any time i, it provides REP(LPF) in constant time,
where LPF is the longest previous factor, i.e. the greedy phrase, a reference
to the list of REP({set of prefixes of LPF}) in constant time and REP(p) in
time O(|p| log log n) for any given pattern p
CiNCT: Compression and retrieval for massive vehicular trajectories via relative movement labeling
In this paper, we present a compressed data structure for moving object
trajectories in a road network, which are represented as sequences of road
edges. Unlike existing compression methods for trajectories in a network, our
method supports pattern matching and decompression from an arbitrary position
while retaining a high compressibility with theoretical guarantees.
Specifically, our method is based on FM-index, a fast and compact data
structure for pattern matching. To enhance the compression, we incorporate the
sparsity of road networks into the data structure. In particular, we present
the novel concepts of relative movement labeling and PseudoRank, each
contributing to significant reductions in data size and query processing time.
Our theoretical analysis and experimental studies reveal the advantages of our
proposed method as compared to existing trajectory compression methods and
FM-index variants
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