5,062 research outputs found
Fast Algorithm for Partial Covers in Words
A factor of a word is a cover of if every position in lies
within some occurrence of in . A word covered by thus
generalizes the idea of a repetition, that is, a word composed of exact
concatenations of . In this article we introduce a new notion of
-partial cover, which can be viewed as a relaxed variant of cover, that
is, a factor covering at least positions in . We develop a data
structure of size (where ) that can be constructed in time which we apply to compute all shortest -partial covers for a
given . We also employ it for an -time algorithm computing
a shortest -partial cover for each
Efficient Seeds Computation Revisited
The notion of the cover is a generalization of a period of a string, and
there are linear time algorithms for finding the shortest cover. The seed is a
more complicated generalization of periodicity, it is a cover of a superstring
of a given string, and the shortest seed problem is of much higher algorithmic
difficulty. The problem is not well understood, no linear time algorithm is
known. In the paper we give linear time algorithms for some of its versions ---
computing shortest left-seed array, longest left-seed array and checking for
seeds of a given length. The algorithm for the last problem is used to compute
the seed array of a string (i.e., the shortest seeds for all the prefixes of
the string) in time. We describe also a simpler alternative algorithm
computing efficiently the shortest seeds. As a by-product we obtain an
time algorithm checking if the shortest seed has length at
least and finding the corresponding seed. We also correct some important
details missing in the previously known shortest-seed algorithm (Iliopoulos et
al., 1996).Comment: 14 pages, accepted to CPM 201
Internal Quasiperiod Queries
Internal pattern matching requires one to answer queries about factors of a
given string. Many results are known on answering internal period queries,
asking for the periods of a given factor. In this paper we investigate (for the
first time) internal queries asking for covers (also known as quasiperiods) of
a given factor. We propose a data structure that answers such queries in
time for the shortest cover and in time for a representation of all the covers, after time
and space preprocessing.Comment: To appear in the SPIRE 2020 proceeding
Searching of gapped repeats and subrepetitions in a word
A gapped repeat is a factor of the form where and are nonempty
words. The period of the gapped repeat is defined as . The gapped
repeat is maximal if it cannot be extended to the left or to the right by at
least one letter with preserving its period. The gapped repeat is called
-gapped if its period is not greater than . A
-subrepetition is a factor which exponent is less than 2 but is not
less than (the exponent of the factor is the quotient of the length
and the minimal period of the factor). The -subrepetition is maximal if
it cannot be extended to the left or to the right by at least one letter with
preserving its minimal period. We reveal a close relation between maximal
gapped repeats and maximal subrepetitions. Moreover, we show that in a word of
length the number of maximal -gapped repeats is bounded by
and the number of maximal -subrepetitions is bounded by
. Using the obtained upper bounds, we propose algorithms for
finding all maximal -gapped repeats and all maximal
-subrepetitions in a word of length . The algorithm for finding all
maximal -gapped repeats has time complexity for the case
of constant alphabet size and time complexity for the
general case. For finding all maximal -subrepetitions we propose two
algorithms. The first algorithm has time
complexity for the case of constant alphabet size and time complexity for the general case. The
second algorithm has
expected time complexity
A_{n-1} singularities and nKdV hierarchies
According to a conjecture of E. Witten proved by M. Kontsevich, a certain
generating function for intersection indices on the Deligne -- Mumford moduli
spaces of Riemann surfaces coincides with a certain tau-function of the KdV
hierarchy. The generating function is naturally generalized under the name the
{\em total descendent potential} in the theory of Gromov -- Witten invariants
of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive:
math.DG/0108160 contain two equivalent constructions, motivated by some results
in Gromov -- Witten theory, which associate a total descendent potential to any
semisimple Frobenius structure. In this paper, we prove that in the case of
K.Saito's Frobenius structure on the miniversal deformation of the
-singularity, the total descendent potential is a tau-function of the
KdV hierarchy. We derive this result from a more general construction for
solutions of the KdV hierarchy from solutions of the KdV hierarchy.Comment: 29 pages, to appear in Moscow Mathematical Journa
Strings on Celestial Sphere
We transform superstring scattering amplitudes into the correlation functions
of primary conformal fields on two-dimensional celestial sphere. The points on
celestial sphere are associated to the asymptotic directions of (light-like)
momenta of external particles, with the Lorentz group realized as the SL(2,C)
conformal symmetry of the sphere. The energies are dualized through Mellin
transforms into the parameters that determine dimensions of the primaries. We
focus on four-point amplitudes involving gauge bosons and gravitons in type I
open superstring theory and in closed heterotic superstring theory at the
tree-level.Comment: 28 pages, harvmac; v2: added Appendix A, final version to appear in
Nucl. Phys.
The Number of Repetitions in 2D-Strings
The notions of periodicity and repetitions in strings, and hence these of
runs and squares, naturally extend to two-dimensional strings. We consider two
types of repetitions in 2D-strings: 2D-runs and quartics (quartics are a
2D-version of squares in standard strings). Amir et al. introduced 2D-runs,
showed that there are of them in an 2D-string and
presented a simple construction giving a lower bound of for their
number (TCS 2020). We make a significant step towards closing the gap between
these bounds by showing that the number of 2D-runs in an 2D-string
is . In particular, our bound implies that the run-time of the algorithm of Amir et al. for computing
2D-runs is also . We expect this result to allow for
exploiting 2D-runs algorithmically in the area of 2D pattern matching.
A quartic is a 2D-string composed of identical blocks
(2D-strings) that was introduced by Apostolico and Brimkov (TCS 2000), where by
quartics they meant only primitively rooted quartics, i.e. built of a primitive
block. Here our notion of quartics is more general and analogous to that of
squares in 1D-strings. Apostolico and Brimkov showed that there are occurrences of primitively rooted quartics in an
2D-string and that this bound is attainable. Consequently the number of
distinct primitively rooted quartics is . Here, we prove that
the number of distinct general quartics is also . This extends
the rich combinatorial study of the number of distinct squares in a 1D-string,
that was initiated by Fraenkel and Simpson (J. Comb. Theory A 1998), to two
dimensions.
Finally, we show some algorithmic applications of 2D-runs. (Abstract
shortened due to arXiv requirements.)Comment: To appear in the ESA 2020 proceeding
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