4,675 research outputs found
Wavelet Trees Meet Suffix Trees
We present an improved wavelet tree construction algorithm and discuss its
applications to a number of rank/select problems for integer keys and strings.
Given a string of length n over an alphabet of size , our
method builds the wavelet tree in time,
improving upon the state-of-the-art algorithm by a factor of .
As a consequence, given an array of n integers we can construct in time a data structure consisting of machine words and
capable of answering rank/select queries for the subranges of the array in
time. This is a -factor improvement in
query time compared to Chan and P\u{a}tra\c{s}cu and a -factor
improvement in construction time compared to Brodal et al.
Next, we switch to stringological context and propose a novel notion of
wavelet suffix trees. For a string w of length n, this data structure occupies
words, takes time to construct, and simultaneously
captures the combinatorial structure of substrings of w while enabling
efficient top-down traversal and binary search. In particular, with a wavelet
suffix tree we are able to answer in time the following two
natural analogues of rank/select queries for suffixes of substrings: for
substrings x and y of w count the number of suffixes of x that are
lexicographically smaller than y, and for a substring x of w and an integer k,
find the k-th lexicographically smallest suffix of x.
We further show that wavelet suffix trees allow to compute a
run-length-encoded Burrows-Wheeler transform of a substring x of w in time, where s denotes the length of the resulting run-length encoding.
This answers a question by Cormode and Muthukrishnan, who considered an
analogous problem for Lempel-Ziv compression.Comment: 33 pages, 5 figures; preliminary version published at SODA 201
Online Bin Stretching with Three Bins
Online Bin Stretching is a semi-online variant of bin packing in which the
algorithm has to use the same number of bins as an optimal packing, but is
allowed to slightly overpack the bins. The goal is to minimize the amount of
overpacking, i.e., the maximum size packed into any bin.
We give an algorithm for Online Bin Stretching with a stretching factor of
for three bins. Additionally, we present a lower bound of for Online Bin Stretching on three bins and a lower bound of
for four and five bins that were discovered using a computer search.Comment: Preprint of a journal version. See version 2 for the conference
paper. Conference paper split into two journal submissions; see
arXiv:1601.0811
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
. (...continues...
Glassy phases in Random Heteropolymers with correlated sequences
We develop a new analytic approach for the study of lattice heteropolymers,
and apply it to copolymers with correlated Markovian sequences. According to
our analysis, heteropolymers present three different dense phases depending
upon the temperature, the nature of the monomer interactions, and the sequence
correlations: (i) a liquid phase, (ii) a ``soft glass'' phase, and (iii) a
``frozen glass'' phase. The presence of the new intermediate ``soft glass''
phase is predicted for instance in the case of polyampholytes with sequences
that favor the alternation of monomers.
Our approach is based on the cavity method, a refined Bethe Peierls
approximation adapted to frustrated systems. It amounts to a mean field
treatment in which the nearest neighbor correlations, which are crucial in the
dense phases of heteropolymers, are handled exactly. This approach is powerful
and versatile, it can be improved systematically and generalized to other
polymeric systems
Top Tree Compression of Tries
We present a compressed representation of tries based on top tree compression [ICALP 2013] that works on a standard, comparison-based, pointer machine model of computation and supports efficient prefix search queries. Namely, we show how to preprocess a set of strings of total length n over an alphabet of size sigma into a compressed data structure of worst-case optimal size O(n/log_sigma n) that given a pattern string P of length m determines if P is a prefix of one of the strings in time O(min(m log sigma,m + log n)). We show that this query time is in fact optimal regardless of the size of the data structure.
Existing solutions either use Omega(n) space or rely on word RAM techniques, such as tabulation, hashing, address arithmetic, or word-level parallelism, and hence do not work on a pointer machine. Our result is the first solution on a pointer machine that achieves worst-case o(n) space. Along the way, we develop several interesting data structures that work on a pointer machine and are of independent interest. These include an optimal data structures for random access to a grammar-compressed string and an optimal data structure for a variant of the level ancestor problem
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Engineering Predecessor Data Structures for Dynamic Integer Sets
We present highly optimized data structures for the dynamic predecessor problem, where the task is to maintain a set S of w-bit numbers under insertions, deletions, and predecessor queries (return the largest element in S no larger than a given key). The problem of finding predecessors can be viewed as a generalized form of the membership problem, or as a simple version of the nearest neighbour problem. It lies at the core of various real-world problems such as internet routing.
In this work, we engineer (1) a simple implementation of the idea of universe reduction, similar to van-Emde-Boas trees (2) variants of y-fast tries [Willard, IPL\u2783], and (3) B-trees with different strategies for organizing the keys contained in the nodes, including an implementation of dynamic fusion nodes [P?tra?cu and Thorup, FOCS\u2714]. We implement our data structures for w = 32,40,64, which covers most typical scenarios.
Our data structures finish workloads faster than previous approaches while being significantly more space-efficient, e.g., they clearly outperform standard implementations of the STL by finishing up to four times as fast using less than a third of the memory. Our tests also provide more general insights on data structure design, such as how small sets should be stored and handled and if and when new CPU instructions such as advanced vector extensions pay off
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