264 research outputs found
Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas
We show lower bounds of and on the
randomized and quantum communication complexity, respectively, of all
-variable read-once Boolean formulas. Our results complement the recent
lower bound of by Leonardos and Saks and
by Jayram, Kopparty and Raghavendra for
randomized communication complexity of read-once Boolean formulas with depth
. We obtain our result by "embedding" either the Disjointness problem or its
complement in any given read-once Boolean formula.Comment: 5 page
Counting flags in triangle-free digraphs
Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph
on n vertices with minimum outdegree 0.3465n contains an oriented triangle.
This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main
new tool we use in our proof is the theory of flag algebras developed recently
by Razborov.Comment: 19 pages, 7 figures; this is the final version to appear in
Combinatoric
Powers of sets in free groups
We prove that |A^n| > c_n |A|^{[\frac{n+1}{2}]} for any finite subset A of a
free group if A contains at least two noncommuting elements, where c_n>0 are
constants not depending on A. Simple examples show that the order of these
estimates are the best possible for each n>0.Comment: 3 page
An Improved Interactive Streaming Algorithm for the Distinct Elements Problem
The exact computation of the number of distinct elements (frequency moment
) is a fundamental problem in the study of data streaming algorithms. We
denote the length of the stream by where each symbol is drawn from a
universe of size . While it is well known that the moments can
be approximated by efficient streaming algorithms, it is easy to see that exact
computation of requires space . In previous work, Cormode
et al. therefore considered a model where the data stream is also processed by
a powerful helper, who provides an interactive proof of the result. They gave
such protocols with a polylogarithmic number of rounds of communication between
helper and verifier for all functions in NC. This number of rounds
can quickly make such
protocols impractical.
Cormode et al. also gave a protocol with rounds for the exact
computation of where the space complexity is but the total communication . They managed to give round protocols with
complexity for many other interesting problems
including , Inner product, and Range-sum, but computing exactly with
polylogarithmic space and communication and rounds remained open.
In this work, we give a streaming interactive protocol with rounds
for exact computation of using bits of space and the communication is . The update
time of the verifier per symbol received is .Comment: Submitted to ICALP 201
On the pigeonhole and related principles in deep inference and monotone systems
International audienceWe construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs. We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs. As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct n^O(log log n) -size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1/ polylog n, thereby also improving the best known bounds for monotone proofs
Circuit Complexity Meets Ontology-Based Data Access
Ontology-based data access is an approach to organizing access to a database
augmented with a logical theory. In this approach query answering proceeds
through a reformulation of a given query into a new one which can be answered
without any use of theory. Thus the problem reduces to the standard database
setting.
However, the size of the query may increase substantially during the
reformulation. In this survey we review a recently developed framework on
proving lower and upper bounds on the size of this reformulation by employing
methods and results from Boolean circuit complexity.Comment: To appear in proceedings of CSR 2015, LNCS 9139, Springe
Generalised state spaces and non-locality in fault tolerant quantum computing schemes
We develop connections between generalised notions of entanglement and
quantum computational devices where the measurements available are restricted,
either because they are noisy and/or because by design they are only along
Pauli directions. By considering restricted measurements one can (by
considering the dual positive operators) construct single particle state spaces
that are different to the usual quantum state space. This leads to a modified
notion of entanglement that can be very different to the quantum version (for
example, Bell states can become separable). We use this approach to develop
alternative methods of classical simulation that have strong connections to the
study of non-local correlations: we construct noisy quantum computers that
admit operations outside the Clifford set and can generate some forms of
multiparty quantum entanglement, but are otherwise classical in that they can
be efficiently simulated classically and cannot generate non-local statistics.
Although the approach provides new regimes of noisy quantum evolution that can
be efficiently simulated classically, it does not appear to lead to significant
reductions of existing upper bounds to fault tolerance thresholds for common
noise models.Comment: V2: 18 sides, 7 figures. Corrected two erroneous claims and one
erroneous argumen
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