2,985 research outputs found
Finding Frequent Subsequences in a Set of Texts
Given a set of strings, the Common Subsequence Automaton accepts all common subsequences of these strings. Such an automaton can be deduced from other automata like the Directed Acyclic Subsequence Graph or the Subsequence Automaton. In this paper, we introduce some new issues in text algorithm on the basis of Common Subsequences related problems. Firstly, we make an overview of different existing automata, focusing on their similarities and differences. Secondly, we present a new automaton, the Constrained Subsequence Automaton, which extends the Common Subsequence Automaton, by adding an integer denoted quorum
Subsequence Automata with Default Transitions
Let be a string of length with characters from an alphabet of size
. The \emph{subsequence automaton} of (often called the
\emph{directed acyclic subsequence graph}) is the minimal deterministic finite
automaton accepting all subsequences of . A straightforward construction
shows that the size (number of states and transitions) of the subsequence
automaton is and that this bound is asymptotically optimal.
In this paper, we consider subsequence automata with \emph{default
transitions}, that is, special transitions to be taken only if none of the
regular transitions match the current character, and which do not consume the
current character. We show that with default transitions, much smaller
subsequence automata are possible, and provide a full trade-off between the
size of the automaton and the \emph{delay}, i.e., the maximum number of
consecutive default transitions followed before consuming a character.
Specifically, given any integer parameter , , we
present a subsequence automaton with default transitions of size
and delay . Hence, with we
obtain an automaton of size and delay . On
the other extreme, with , we obtain an automaton of size and delay , thus matching the bound for the standard subsequence
automaton construction. Finally, we generalize the result to multiple strings.
The key component of our result is a novel hierarchical automata construction
of independent interest.Comment: Corrected typo
Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree
A heapable sequence is a sequence of numbers that can be arranged in a min-heap data structure. Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence and maximum-sized binary tree. We introduce alphabet size as a new parameter in the study of computational problems in permutation DAGs and show that this parameter with respect to a fixed topological ordering admits a complete characterization and a polynomial time algorithm. We believe that this parameter is likely to be useful in the context of optimization problems defined over permutation DAGs
Improving legibility of natural deduction proofs is not trivial
In formal proof checking environments such as Mizar it is not merely the
validity of mathematical formulas that is evaluated in the process of adoption
to the body of accepted formalizations, but also the readability of the proofs
that witness validity. As in case of computer programs, such proof scripts may
sometimes be more and sometimes be less readable. To better understand the
notion of readability of formal proofs, and to assess and improve their
readability, we propose in this paper a method of improving proof readability
based on Behaghel's First Law of sentence structure. Our method maximizes the
number of local references to the directly preceding statement in a proof
linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
3D Visibility Representations of 1-planar Graphs
We prove that every 1-planar graph G has a z-parallel visibility
representation, i.e., a 3D visibility representation in which the vertices are
isothetic disjoint rectangles parallel to the xy-plane, and the edges are
unobstructed z-parallel visibilities between pairs of rectangles. In addition,
the constructed representation is such that there is a plane that intersects
all the rectangles, and this intersection defines a bar 1-visibility
representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Two novel evolutionary formulations of the graph coloring problem
We introduce two novel evolutionary formulations of the problem of coloring
the nodes of a graph. The first formulation is based on the relationship that
exists between a graph's chromatic number and its acyclic orientations. It
views such orientations as individuals and evolves them with the aid of
evolutionary operators that are very heavily based on the structure of the
graph and its acyclic orientations. The second formulation, unlike the first
one, does not tackle one graph at a time, but rather aims at evolving a
`program' to color all graphs belonging to a class whose members all have the
same number of nodes and other common attributes. The heuristics that result
from these formulations have been tested on some of the Second DIMACS
Implementation Challenge benchmark graphs, and have been found to be
competitive when compared to the several other heuristics that have also been
tested on those graphs.Comment: To appear in Journal of Combinatorial Optimizatio
Improving the smoothed complexity of FLIP for max cut problems
Finding locally optimal solutions for max-cut and max--cut are well-known
PLS-complete problems. An instinctive approach to finding such a locally
optimum solution is the FLIP method. Even though FLIP requires exponential time
in worst-case instances, it tends to terminate quickly in practical instances.
To explain this discrepancy, the run-time of FLIP has been studied in the
smoothed complexity framework. Etscheid and R\"{o}glin showed that the smoothed
complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel,
Bubeck, Peres, and Wei showed that the smoothed complexity of FLIP for max-cut
in complete graphs is , where is an upper bound on
the random edge-weight density and is the number of vertices in the input
graph.
While Angel et al.'s result showed the first polynomial smoothed complexity,
they also conjectured that their run-time bound is far from optimal. In this
work, we make substantial progress towards improving the run-time bound. We
prove that the smoothed complexity of FLIP in complete graphs is . Our results are based on a carefully chosen matrix whose rank
captures the run-time of the method along with improved rank bounds for this
matrix and an improved union bound based on this matrix. In addition, our
techniques provide a general framework for analyzing FLIP in the smoothed
framework. We illustrate this general framework by showing that the smoothed
complexity of FLIP for max--cut in complete graphs is polynomial and for
max--cut in arbitrary graphs is quasi-polynomial. We believe that our
techniques should also be of interest towards addressing the smoothed
complexity of FLIP for max--cut in complete graphs for larger constants .Comment: 36 page
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