5,660 research outputs found
Linear Compressed Pattern Matching for Polynomial Rewriting (Extended Abstract)
This paper is an extended abstract of an analysis of term rewriting where the
terms in the rewrite rules as well as the term to be rewritten are compressed
by a singleton tree grammar (STG). This form of compression is more general
than node sharing or representing terms as dags since also partial trees
(contexts) can be shared in the compression. In the first part efficient but
complex algorithms for detecting applicability of a rewrite rule under
STG-compression are constructed and analyzed. The second part applies these
results to term rewriting sequences.
The main result for submatching is that finding a redex of a left-linear rule
can be performed in polynomial time under STG-compression.
The main implications for rewriting and (single-position or parallel)
rewriting steps are: (i) under STG-compression, n rewriting steps can be
performed in nondeterministic polynomial time. (ii) under STG-compression and
for left-linear rewrite rules a sequence of n rewriting steps can be performed
in polynomial time, and (iii) for compressed rewrite rules where the left hand
sides are either DAG-compressed or ground and STG-compressed, and an
STG-compressed target term, n rewriting steps can be performed in polynomial
time.Comment: In Proceedings TERMGRAPH 2013, arXiv:1302.599
Efficient Computation of Sequence Mappability
Sequence mappability is an important task in genome re-sequencing. In the
-mappability problem, for a given sequence of length , our goal
is to compute a table whose th entry is the number of indices such
that length- substrings of starting at positions and have at
most mismatches. Previous works on this problem focused on heuristic
approaches to compute a rough approximation of the result or on the case of
. We present several efficient algorithms for the general case of the
problem. Our main result is an algorithm that works in time and space for
. It requires a carefu l adaptation of the technique of Cole
et al.~[STOC 2004] to avoid multiple counting of pairs of substrings. We also
show -time algorithms to compute all results for a fixed
and all or a fixed and all . Finally we show
that the -mappability problem cannot be solved in strongly subquadratic
time for unless the Strong Exponential Time Hypothesis
fails.Comment: Accepted to SPIRE 201
Faster Approximate String Matching for Short Patterns
We study the classical approximate string matching problem, that is, given
strings and and an error threshold , find all ending positions of
substrings of whose edit distance to is at most . Let and
have lengths and , respectively. On a standard unit-cost word RAM with
word size we present an algorithm using time When is
short, namely, or this
improves the previously best known time bounds for the problem. The result is
achieved using a novel implementation of the Landau-Vishkin algorithm based on
tabulation and word-level parallelism.Comment: To appear in Theory of Computing System
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