3,577 research outputs found
Fully dynamic data structure for LCE queries in compressed space
A Longest Common Extension (LCE) query on a text of length asks for
the length of the longest common prefix of suffixes starting at given two
positions. We show that the signature encoding of size [Mehlhorn et al., Algorithmica 17(2):183-198,
1997] of , which can be seen as a compressed representation of , has a
capability to support LCE queries in time,
where is the answer to the query, is the size of the Lempel-Ziv77
(LZ77) factorization of , and is an integer that can be handled
in constant time under word RAM model. In compressed space, this is the fastest
deterministic LCE data structure in many cases. Moreover, can be
enhanced to support efficient update operations: After processing
in time, we can insert/delete any (sub)string of length
into/from an arbitrary position of in time, where . This yields
the first fully dynamic LCE data structure. We also present efficient
construction algorithms from various types of inputs: We can construct
in time from uncompressed string ; in
time from grammar-compressed string
represented by a straight-line program of size ; and in time from LZ77-compressed string with factors. On top
of the above contributions, we show several applications of our data structures
which improve previous best known results on grammar-compressed string
processing.Comment: arXiv admin note: text overlap with arXiv:1504.0695
c-trie++: A Dynamic Trie Tailored for Fast Prefix Searches
Given a dynamic set of strings of total length whose characters
are drawn from an alphabet of size , a keyword dictionary is a data
structure built on that provides locate, prefix search, and update
operations on . Under the assumption that
characters fit into a single machine word , we propose a keyword dictionary
that represents in bits of space,
supporting all operations in expected time on an
input string of length in the word RAM model. This data structure is
underlined with an exhaustive practical evaluation, highlighting the practical
usefulness of the proposed data structure, especially for prefix searches - one
of the most elementary keyword dictionary operations
Repetition Detection in a Dynamic String
A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings.
We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting
Faster Compact On-Line Lempel-Ziv Factorization
We present a new on-line algorithm for computing the Lempel-Ziv factorization
of a string that runs in time and uses only bits
of working space, where is the length of the string and is the
size of the alphabet. This is a notable improvement compared to the performance
of previous on-line algorithms using the same order of working space but
running in either time (Okanohara & Sadakane 2009) or
time (Starikovskaya 2012). The key to our new algorithm is in the
utilization of an elegant but less popular index structure called Directed
Acyclic Word Graphs, or DAWGs (Blumer et al. 1985). We also present an
opportunistic variant of our algorithm, which, given the run length encoding of
size of a string of length , computes the Lempel-Ziv factorization
on-line, in time
and bits of space, which is faster and more space efficient when
the string is run-length compressible
Dynamic Relative Compression, Dynamic Partial Sums, and Substring Concatenation
Given a static reference string and a source string , a relative
compression of with respect to is an encoding of as a sequence of
references to substrings of . Relative compression schemes are a classic
model of compression and have recently proved very successful for compressing
highly-repetitive massive data sets such as genomes and web-data. We initiate
the study of relative compression in a dynamic setting where the compressed
source string is subject to edit operations. The goal is to maintain the
compressed representation compactly, while supporting edits and allowing
efficient random access to the (uncompressed) source string. We present new
data structures that achieve optimal time for updates and queries while using
space linear in the size of the optimal relative compression, for nearly all
combinations of parameters. We also present solutions for restricted and
extended sets of updates. To achieve these results, we revisit the dynamic
partial sums problem and the substring concatenation problem. We present new
optimal or near optimal bounds for these problems. Plugging in our new results
we also immediately obtain new bounds for the string indexing for patterns with
wildcards problem and the dynamic text and static pattern matching problem
The Wavelet Trie: Maintaining an Indexed Sequence of Strings in Compressed Space
An indexed sequence of strings is a data structure for storing a string
sequence that supports random access, searching, range counting and analytics
operations, both for exact matches and prefix search. String sequences lie at
the core of column-oriented databases, log processing, and other storage and
query tasks. In these applications each string can appear several times and the
order of the strings in the sequence is relevant. The prefix structure of the
strings is relevant as well: common prefixes are sought in strings to extract
interesting features from the sequence. Moreover, space-efficiency is highly
desirable as it translates directly into higher performance, since more data
can fit in fast memory.
We introduce and study the problem of compressed indexed sequence of strings,
representing indexed sequences of strings in nearly-optimal compressed space,
both in the static and dynamic settings, while preserving provably good
performance for the supported operations.
We present a new data structure for this problem, the Wavelet Trie, which
combines the classical Patricia Trie with the Wavelet Tree, a succinct data
structure for storing a compressed sequence. The resulting Wavelet Trie
smoothly adapts to a sequence of strings that changes over time. It improves on
the state-of-the-art compressed data structures by supporting a dynamic
alphabet (i.e. the set of distinct strings) and prefix queries, both crucial
requirements in the aforementioned applications, and on traditional indexes by
reducing space occupancy to close to the entropy of the sequence
Managing Unbounded-Length Keys in Comparison-Driven Data Structures with Applications to On-Line Indexing
This paper presents a general technique for optimally transforming any
dynamic data structure that operates on atomic and indivisible keys by
constant-time comparisons, into a data structure that handles unbounded-length
keys whose comparison cost is not a constant. Examples of these keys are
strings, multi-dimensional points, multiple-precision numbers, multi-key data
(e.g.~records), XML paths, URL addresses, etc. The technique is more general
than what has been done in previous work as no particular exploitation of the
underlying structure of is required. The only requirement is that the insertion
of a key must identify its predecessor or its successor.
Using the proposed technique, online suffix tree can be constructed in worst
case time per input symbol (as opposed to amortized
time per symbol, achieved by previously known algorithms). To our knowledge,
our algorithm is the first that achieves worst case time per input
symbol. Searching for a pattern of length in the resulting suffix tree
takes time, where is the
number of occurrences of the pattern. The paper also describes more
applications and show how to obtain alternative methods for dealing with suffix
sorting, dynamic lowest common ancestors and order maintenance
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