334,628 research outputs found
Boolean Operations, Joins, and the Extended Low Hierarchy
We prove that the join of two sets may actually fall into a lower level of
the extended low hierarchy than either of the sets. In particular, there exist
sets that are not in the second level of the extended low hierarchy, EL_2, yet
their join is in EL_2. That is, in terms of extended lowness, the join operator
can lower complexity. Since in a strong intuitive sense the join does not lower
complexity, our result suggests that the extended low hierarchy is unnatural as
a complexity measure. We also study the closure properties of EL_ and prove
that EL_2 is not closed under certain Boolean operations. To this end, we
establish the first known (and optimal) EL_2 lower bounds for certain notions
generalizing Selman's P-selectivity, which may be regarded as an interesting
result in its own right.Comment: 12 page
Joins of DGA modules and sectional category
We construct an explicit semifree model for the fiber join of two fibrations
p: E --> B and p': E' --> B from semifree models of p and p'. Using this model,
we introduce a lower bound of the sectional category of a fibration p which can
be calculated from any Sullivan model of p and which is closer to the sectional
category of p than the classical cohomological lower bound given by the
nilpotency of the kernel of p^*: H^*(B;Q) --> H^*(E;Q). In the special case of
the evaluation fibration X^I --> X x X we obtain a computable lower bound of
Farber's topological complexity TC(X). We show that the difference between this
lower bound and the classical cohomological lower bound can be arbitrarily
large.Comment: This is the version published by Algebraic & Geometric Topology on 24
February 200
Counting and Computing Join-Endomorphisms in Lattices (Revisited)
Structures involving a lattice and join-endomorphisms on it are ubiquitous in
computer science. We study the cardinality of the set of all
join-endomorphisms of a given finite lattice . In particular, we show for
, the discrete order of elements extended with top and
bottom, where
is the Laguerre polynomial of degree . We also study the
following problem: Given a lattice of size and a set of size , find the greatest lower bound
. The join-endomorphism
has meaningful interpretations in epistemic
logic, distributed systems, and Aumann structures. We show that this problem
can be solved with worst-case time complexity in for distributive
lattices and for arbitrary lattices. In the particular case of
modular lattices, we present an adaptation of the latter algorithm that reduces
its average time complexity. We provide theoretical and experimental results to
support this enhancement. The complexity is expressed in terms of the basic
binary lattice operations performed by the algorithm
Worst-Case Optimal Algorithms for Parallel Query Processing
In this paper, we study the communication complexity for the problem of
computing a conjunctive query on a large database in a parallel setting with
servers. In contrast to previous work, where upper and lower bounds on the
communication were specified for particular structures of data (either data
without skew, or data with specific types of skew), in this work we focus on
worst-case analysis of the communication cost. The goal is to find worst-case
optimal parallel algorithms, similar to the work of [18] for sequential
algorithms.
We first show that for a single round we can obtain an optimal worst-case
algorithm. The optimal load for a conjunctive query when all relations have
size equal to is , where is a new query-related
quantity called the edge quasi-packing number, which is different from both the
edge packing number and edge cover number of the query hypergraph. For multiple
rounds, we present algorithms that are optimal for several classes of queries.
Finally, we show a surprising connection to the external memory model, which
allows us to translate parallel algorithms to external memory algorithms. This
technique allows us to recover (within a polylogarithmic factor) several recent
results on the I/O complexity for computing join queries, and also obtain
optimal algorithms for other classes of queries
Space-Efficient Data Structures for Lattices
A lattice is a partially-ordered set in which every pair of elements has a
unique meet (greatest lower bound) and join (least upper bound). We present new
data structures for lattices that are simple, efficient, and nearly optimal in
terms of space complexity.
Our first data structure can answer partial order queries in constant time
and find the meet or join of two elements in time, where is
the number of elements in the lattice. It occupies bits of
space, which is only a factor from the -bit
lower bound for storing lattices. The preprocessing time is . This
structure admits a simple space-time tradeoff so that, for any , the data structure supports meet and join queries in
time, occupies bits of space, and can be
constructed in time.
Our second data structure uses bits of space and supports
meet and join in time, where is the maximum
degree of any element in the transitive reduction graph of the lattice. This
structure is much faster for lattices with low-degree elements.
This paper also identifies an error in a long-standing solution to the
problem of representing lattices. We discuss the issue with this previous work.Comment: Accepted in SWAT 202
Conjunctive Queries on Probabilistic Graphs: The Limits of Approximability
Query evaluation over probabilistic databases is a notoriously intractable
problem -- not only in combined complexity, but for many natural queries in
data complexity as well. This motivates the study of probabilistic query
evaluation through the lens of approximation algorithms, and particularly of
combined FPRASes, whose runtime is polynomial in both the query and instance
size. In this paper, we focus on tuple-independent probabilistic databases over
binary signatures, which can be equivalently viewed as probabilistic graphs. We
study in which cases we can devise combined FPRASes for probabilistic query
evaluation in this setting.
We settle the complexity of this problem for a variety of query and instance
classes, by proving both approximability and (conditional) inapproximability
results. This allows us to deduce many corollaries of possible independent
interest. For example, we show how the results of Arenas et al. on counting
fixed-length strings accepted by an NFA imply the existence of an FPRAS for the
two-terminal network reliability problem on directed acyclic graphs: this was
an open problem until now. We also show that one cannot extend the recent
result of van Bremen and Meel that gives a combined FPRAS for self-join-free
conjunctive queries of bounded hypertree width on probabilistic databases:
neither the bounded-hypertree-width condition nor the self-join-freeness
hypothesis can be relaxed. Finally, we complement all our inapproximability
results with unconditional lower bounds, showing that DNNF provenance circuits
must have at least moderately exponential size in combined complexity.Comment: 19 pages. Submitte
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
Guaranteeing the \~O(AGM/OUT) Runtime for Uniform Sampling and OUT Size Estimation over Joins
We propose a new method for estimating the number of answers OUT of a small
join query Q in a large database D, and for uniform sampling over joins. Our
method is the first to satisfy all the following statements. - Support
arbitrary Q, which can be either acyclic or cyclic, and contain binary and
non-binary relations. - Guarantee an arbitrary small error with a high
probability always in \~O(AGM/OUT) time, where AGM is the AGM bound OUT (an
upper bound of OUT), and \~O hides the polylogarithmic factor of input size. We
also explain previous join size estimators in a unified framework. All methods
including ours rely on certain indexes on relations in D, which take linear
time to build offline. Additionally, we extend our method using generalized
hypertree decompositions (GHDs) to achieve a lower complexity than \~O(AGM/OUT)
when OUT is small, and present optimization techniques for improving estimation
efficiency and accuracy.Comment: 19 page
Quantum Communication Complexity of Distributed Set Joins
Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the complexity of such problems in the natural model of (classical) two-party communication complexity and obtained tight bounds for the complexity of several important distributed set joins.
In this paper we initiate the study of the quantum communication complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two n times n Boolean matrices, each owned by one of two respective parties, can be computed with widetilde-O(sqrt{n} ell^{3/4}) qubits of communication, where ell denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical communication complexity of this problem is widetilde-Theta(n sqrt{ell}), our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the communication complexity of distributed matrix multiplication over F_2.
Besides their applications to database theory, the communication complexity of set joins is interesting due to its connections to direct product theorems in communication complexity. In this work we also introduce a notion of all-pairs product theorem, and relate this notion to standard direct product theorems in communication complexity
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