56,059 research outputs found
On insertion-deletion systems over relational words
We introduce a new notion of a relational word as a finite totally ordered
set of positions endowed with three binary relations that describe which
positions are labeled by equal data, by unequal data and those having an
undefined relation between their labels. We define the operations of insertion
and deletion on relational words generalizing corresponding operations on
strings. We prove that the transitive and reflexive closure of these operations
has a decidable membership problem for the case of short insertion-deletion
rules (of size two/three and three/two). At the same time, we show that in the
general case such systems can produce a coding of any recursively enumerable
language leading to undecidabilty of reachability questions.Comment: 24 pages, 8 figure
Sketch-based Randomized Algorithms for Dynamic Graph Regression
A well-known problem in data science and machine learning is {\em linear
regression}, which is recently extended to dynamic graphs. Existing exact
algorithms for updating the solution of dynamic graph regression problem
require at least a linear time (in terms of : the size of the graph).
However, this time complexity might be intractable in practice. In the current
paper, we utilize {\em subsampled randomized Hadamard transform} and
\textsf{CountSketch} to propose the first randomized algorithms. Suppose that
we are given an matrix embedding of the graph, where .
Let be the number of samples required for a guaranteed approximation error,
which is a sublinear function of . Our first algorithm reduces time
complexity of pre-processing to .
Then after an edge insertion or an edge deletion, it updates the approximate
solution in time. Our second algorithm reduces time complexity of
pre-processing to , where is the number of nonzero elements of . Then after
an edge insertion or an edge deletion or a node insertion or a node deletion,
it updates the approximate solution in time, with
. Finally, we show
that under some assumptions, if our first algorithm
outperforms our second algorithm and if our second
algorithm outperforms our first algorithm
Input-Driven Tissue P Automata
We introduce several variants of input-driven tissue P automata where the
rules to be applied only depend on the input symbol. Both strings and multisets are
considered as input objects; the strings are either read from an input tape or defined
by the sequence of symbols taken in, and the multisets are given in an input cell at the
beginning of a computation, enclosed in a vesicle. Additional symbols generated during a
computation are stored in this vesicle, too. An input is accepted when the vesicle reaches a
final cell and it is empty. The computational power of some variants of input-driven tissue
P automata is illustrated by examples and compared with the power of the input-driven
variants of other automata as register machines and counter automata
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
We develop data structures for dynamic closest pair problems with arbitrary
distance functions, that do not necessarily come from any geometric structure
on the objects. Based on a technique previously used by the author for
Euclidean closest pairs, we show how to insert and delete objects from an
n-object set, maintaining the closest pair, in O(n log^2 n) time per update and
O(n) space. With quadratic space, we can instead use a quadtree-like structure
to achieve an optimal time bound, O(n) per update. We apply these data
structures to hierarchical clustering, greedy matching, and TSP heuristics, and
discuss other potential applications in machine learning, Groebner bases, and
local improvement algorithms for partition and placement problems. Experiments
show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at
the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp.
619-628. For source code and experimental results, see
http://www.ics.uci.edu/~eppstein/projects/pairs
(Tissue) P Systems with Vesicles of Multisets
We consider tissue P systems working on vesicles of multisets with the very
simple operations of insertion, deletion, and substitution of single objects.
With the whole multiset being enclosed in a vesicle, sending it to a target
cell can be indicated in those simple rules working on the multiset. As
derivation modes we consider the sequential mode, where exactly one rule is
applied in a derivation step, and the set maximal mode, where in each
derivation step a non-extendable set of rules is applied. With the set maximal
mode, computational completeness can already be obtained with tissue P systems
having a tree structure, whereas tissue P systems even with an arbitrary
communication structure are not computationally complete when working in the
sequential mode. Adding polarizations (-1, 0, 1 are sufficient) allows for
obtaining computational completeness even for tissue P systems working in the
sequential mode.Comment: In Proceedings AFL 2017, arXiv:1708.0622
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