104,907 research outputs found
A Static Optimality Transformation with Applications to Planar Point Location
Over the last decade, there have been several data structures that, given a
planar subdivision and a probability distribution over the plane, provide a way
for answering point location queries that is fine-tuned for the distribution.
All these methods suffer from the requirement that the query distribution must
be known in advance.
We present a new data structure for point location queries in planar
triangulations. Our structure is asymptotically as fast as the optimal
structures, but it requires no prior information about the queries. This is a
2D analogue of the jump from Knuth's optimum binary search trees (discovered in
1971) to the splay trees of Sleator and Tarjan in 1985. While the former need
to know the query distribution, the latter are statically optimal. This means
that we can adapt to the query sequence and achieve the same asymptotic
performance as an optimum static structure, without needing any additional
information.Comment: 13 pages, 1 figure, a preliminary version appeared at SoCG 201
Cover Tree Bayesian Reinforcement Learning
This paper proposes an online tree-based Bayesian approach for reinforcement
learning. For inference, we employ a generalised context tree model. This
defines a distribution on multivariate Gaussian piecewise-linear models, which
can be updated in closed form. The tree structure itself is constructed using
the cover tree method, which remains efficient in high dimensional spaces. We
combine the model with Thompson sampling and approximate dynamic programming to
obtain effective exploration policies in unknown environments. The flexibility
and computational simplicity of the model render it suitable for many
reinforcement learning problems in continuous state spaces. We demonstrate this
in an experimental comparison with least squares policy iteration
Sublinearly space bounded iterative arrays
Iterative arrays (IAs) are a, parallel computational model with a sequential processing of the input. They are one-dimensional arrays of interacting identical deterministic finite automata. In this note, realtime-lAs with sublinear space bounds are used to accept formal languages. The existence of a proper hierarchy of space complexity classes between logarithmic anel linear space bounds is proved. Furthermore, an optimal spacc lower bound for non-regular language recognition is shown. Key words: Iterative arrays, cellular automata, space bounded computations, decidability questions, formal languages, theory of computatio
String Synchronizing Sets: Sublinear-Time BWT Construction and Optimal LCE Data Structure
Burrows-Wheeler transform (BWT) is an invertible text transformation that,
given a text of length , permutes its symbols according to the
lexicographic order of suffixes of . BWT is one of the most heavily studied
algorithms in data compression with numerous applications in indexing, sequence
analysis, and bioinformatics. Its construction is a bottleneck in many
scenarios, and settling the complexity of this task is one of the most
important unsolved problems in sequence analysis that has remained open for 25
years. Given a binary string of length , occupying machine
words, the BWT construction algorithm due to Hon et al. (SIAM J. Comput., 2009)
runs in time and space. Recent advancements (Belazzougui,
STOC 2014, and Munro et al., SODA 2017) focus on removing the alphabet-size
dependency in the time complexity, but they still require time.
In this paper, we propose the first algorithm that breaks the -time
barrier for BWT construction. Given a binary string of length , our
procedure builds the Burrows-Wheeler transform in time and
space. We complement this result with a conditional lower bound
proving that any further progress in the time complexity of BWT construction
would yield faster algorithms for the very well studied problem of counting
inversions: it would improve the state-of-the-art -time
solution by Chan and P\v{a}tra\c{s}cu (SODA 2010). Our algorithm is based on a
novel concept of string synchronizing sets, which is of independent interest.
As one of the applications, we show that this technique lets us design a data
structure of the optimal size that answers Longest Common
Extension queries (LCE queries) in time and, furthermore, can be
deterministically constructed in the optimal time.Comment: Full version of a paper accepted to STOC 201
Fully-Functional Suffix Trees and Optimal Text Searching 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 r, the number of runs in their Burrows-Wheeler Transforms
(BWTs). One of the earliest indexes for repetitive collections, the Run-Length
FM-index, used O(r) space and was able to efficiently count the number of
occurrences of a pattern of length m 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
r. In this paper we close this long-standing problem, showing how to extend the
Run-Length FM-index so that it can locate the occ occurrences efficiently
within O(r) space (in loglogarithmic time each), and reaching optimal time, O(m
+ occ), within O(r log log w ({\sigma} + n/r)) space, for a text of length n
over an alphabet of size {\sigma} on a RAM machine with words of w =
{\Omega}(log n) bits. Within that space, our index can also count in optimal
time, O(m). Multiplying the space by O(w/ log {\sigma}), we support count and
locate in O(dm log({\sigma})/we) and O(dm log({\sigma})/we + occ) time, which
is optimal in the packed setting and had not been obtained before in compressed
space. We also describe a structure using O(r log(n/r)) space that replaces the
text and extracts any text substring of length ` in almost-optimal time
O(log(n/r) + ` log({\sigma})/w). Within that space, we similarly provide direct
access to suffix array, inverse suffix array, and longest common prefix array
cells, and extend these capabilities to full suffix tree functionality,
typically in O(log(n/r)) time per operation.Comment: submitted version; optimal count and locate in smaller space: O(r log
log_w(n/r + sigma)
Sequential Design for Optimal Stopping Problems
We propose a new approach to solve optimal stopping problems via simulation.
Working within the backward dynamic programming/Snell envelope framework, we
augment the methodology of Longstaff-Schwartz that focuses on approximating the
stopping strategy. Namely, we introduce adaptive generation of the stochastic
grids anchoring the simulated sample paths of the underlying state process.
This allows for active learning of the classifiers partitioning the state space
into the continuation and stopping regions. To this end, we examine sequential
design schemes that adaptively place new design points close to the stopping
boundaries. We then discuss dynamic regression algorithms that can implement
such recursive estimation and local refinement of the classifiers. The new
algorithm is illustrated with a variety of numerical experiments, showing that
an order of magnitude savings in terms of design size can be achieved. We also
compare with existing benchmarks in the context of pricing multi-dimensional
Bermudan options.Comment: 24 page
A survey of max-type recursive distributional equations
In certain problems in a variety of applied probability settings (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form X =^d
g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are
independent copies of the unknown distribution X. We survey this area,
emphasizing examples where the function g(\cdot) is essentially a ``maximum''
or ``minimum'' function. We draw attention to the theoretical question of
endogeny: in the associated recursive tree process X_i, are the X_i measurable
functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
- …