7 research outputs found
Binary Jumbled String Matching for Highly Run-Length Compressible Texts
The Binary Jumbled String Matching problem is defined as: Given a string
over of length and a query , with non-negative
integers, decide whether has a substring with exactly 's and
's. Previous solutions created an index of size O(n) in a pre-processing
step, which was then used to answer queries in constant time. The fastest
algorithms for construction of this index have running time
[Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or in
the word-RAM model [Moosa and Rahman, JDA 2012]. We propose an index
constructed directly from the run-length encoding of . The construction time
of our index is , where O(n) is the time for computing
the run-length encoding of and is the length of this encoding---this
is no worse than previous solutions if and better if . Our index can be queried in time. While
in the worst case, preliminary investigations have
indicated that may often be close to . Furthermore, the algorithm
for constructing the index is conceptually simple and easy to implement. In an
attempt to shed light on the structure and size of our index, we characterize
it in terms of the prefix normal forms of introduced in [Fici and Lipt\'ak,
DLT 2011].Comment: v2: only small cosmetic changes; v3: new title, weakened conjectures
on size of Corner Index (we no longer conjecture it to be always linear in
size of RLE); removed experimental part on random strings (these are valid
but limited in their predictive power w.r.t. general strings); v3 published
in IP
Algorithms for Computing Abelian Periods of Words
Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the
notion of an \emph{Abelian period} of a word. A word of length over an
alphabet of size can have distinct Abelian periods.
The Brute-Force algorithm computes all the Abelian periods of a word in time
using space. We present an off-line
algorithm based on a \sel function having the same worst-case theoretical
complexity as the Brute-Force one, but outperforming it in practice. We then
present on-line algorithms that also enable to compute all the Abelian periods
of all the prefixes of .Comment: Accepted for publication in Discrete Applied Mathematic
On the Parikh-de-Bruijn grid
We introduce the Parikh-de-Bruijn grid, a graph whose vertices are
fixed-order Parikh vectors, and whose edges are given by a simple shift
operation. This graph gives structural insight into the nature of sets of
Parikh vectors as well as that of the Parikh set of a given string. We show its
utility by proving some results on Parikh-de-Bruijn strings, the abelian analog
of de-Bruijn sequences.Comment: 18 pages, 3 figures, 1 tabl
Reconstruction of Trees from Jumbled and Weighted Subtrees
Let T be an edge-labeled graph, where the labels are from a finite alphabet Sigma. For a subtree U of T the Parikh vector of U is a vector of length |Sigma| which specifies the multiplicity of each label in U. We ask when T can be reconstructed from the multiset of Parikh vectors of all its subtrees, or all of its paths, or all of its maximal paths. We consider the analogous problems for weighted trees. We show how several well-known reconstruction problems on labeled strings, weighted strings and point sets on a line can be included in this framework. We present reconstruction algorithms and non-reconstructibility results, and extend the polynomial method, previously applied to jumbled strings [Acharya et al., SIAM J. on Discr. Math, 2015] and weighted strings [Bansal et al., CPM 2004], to deal with general trees and special tree classes