3,811 research outputs found
Answering Conjunctive Queries under Updates
We consider the task of enumerating and counting answers to -ary
conjunctive queries against relational databases that may be updated by
inserting or deleting tuples. We exhibit a new notion of q-hierarchical
conjunctive queries and show that these can be maintained efficiently in the
following sense. During a linear time preprocessing phase, we can build a data
structure that enables constant delay enumeration of the query results; and
when the database is updated, we can update the data structure and restart the
enumeration phase within constant time. For the special case of self-join free
conjunctive queries we obtain a dichotomy: if a query is not q-hierarchical,
then query enumeration with sublinear delay and sublinear update time
(and arbitrary preprocessing time) is impossible.
For answering Boolean conjunctive queries and for the more general problem of
counting the number of solutions of k-ary queries we obtain complete
dichotomies: if the query's homomorphic core is q-hierarchical, then size of
the the query result can be computed in linear time and maintained with
constant update time. Otherwise, the size of the query result cannot be
maintained with sublinear update time. All our lower bounds rely on the
OMv-conjecture, a conjecture on the hardness of online matrix-vector
multiplication that has recently emerged in the field of fine-grained
complexity to characterise the hardness of dynamic problems. The lower bound
for the counting problem additionally relies on the orthogonal vectors
conjecture, which in turn is implied by the strong exponential time hypothesis.
By sublinear we mean for some
, where is the size of the active domain of the current
database
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
A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM
We present a randomized parallel algorithm that computes the greatest common
divisor of two integers of n bits in length with probability 1-o(1) that takes
O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW
PRAM parallel model of computation. We believe this to be the first randomized
sublinear time algorithm on the EREW PRAM for this problem
- …