47,119 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
Query Learning with Exponential Query Costs
In query learning, the goal is to identify an unknown object while minimizing
the number of "yes" or "no" questions (queries) posed about that object. A
well-studied algorithm for query learning is known as generalized binary search
(GBS). We show that GBS is a greedy algorithm to optimize the expected number
of queries needed to identify the unknown object. We also generalize GBS in two
ways. First, we consider the case where the cost of querying grows
exponentially in the number of queries and the goal is to minimize the expected
exponential cost. Then, we consider the case where the objects are partitioned
into groups, and the objective is to identify only the group to which the
object belongs. We derive algorithms to address these issues in a common,
information-theoretic framework. In particular, we present an exact formula for
the objective function in each case involving Shannon or Renyi entropy, and
develop a greedy algorithm for minimizing it. Our algorithms are demonstrated
on two applications of query learning, active learning and emergency response.Comment: 15 page
Identifiability of Large Phylogenetic Mixture Models
Phylogenetic mixture models are statistical models of character evolution
allowing for heterogeneity. Each of the classes in some unknown partition of
the characters may evolve by different processes, or even along different
trees. The fundamental question of whether parameters of such a model are
identifiable is difficult to address, due to the complexity of the
parameterization. We analyze mixture models on large trees, with many mixture
components, showing that both numerical and tree parameters are indeed
identifiable in these models when all trees are the same. We also explore the
extent to which our algebraic techniques can be employed to extend the result
to mixtures on different trees.Comment: 15 page
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