186,944 research outputs found
Exponential Separation of Quantum and Classical Online Space Complexity
Although quantum algorithms realizing an exponential time speed-up over the
best known classical algorithms exist, no quantum algorithm is known performing
computation using less space resources than classical algorithms. In this
paper, we study, for the first time explicitly, space-bounded quantum
algorithms for computational problems where the input is given not as a whole,
but bit by bit. We show that there exist such problems that a quantum computer
can solve using exponentially less work space than a classical computer. More
precisely, we introduce a very natural and simple model of a space-bounded
quantum online machine and prove an exponential separation of classical and
quantum online space complexity, in the bounded-error setting and for a total
language. The language we consider is inspired by a communication problem (the
set intersection function) that Buhrman, Cleve and Wigderson used to show an
almost quadratic separation of quantum and classical bounded-error
communication complexity. We prove that, in the framework of online space
complexity, the separation becomes exponential.Comment: 13 pages. v3: minor change
Secret Key Agreement from Correlated Data, with No Prior Information
A fundamental question that has been studied in cryptography and in
information theory is whether two parties can communicate confidentially using
exclusively an open channel. We consider the model in which the two parties
hold inputs that are correlated in a certain sense. This model has been studied
extensively in information theory, and communication protocols have been
designed which exploit the correlation to extract from the inputs a shared
secret key. However, all the existing protocols are not universal in the sense
that they require that the two parties also know some attributes of the
correlation. In other words, they require that each party knows something about
the other party's input. We present a protocol that does not require any prior
additional information. It uses space-bounded Kolmogorov complexity to measure
correlation and it allows the two legal parties to obtain a common key that
looks random to an eavesdropper that observes the communication and is
restricted to use a bounded amount of space for the attack. Thus the protocol
achieves complexity-theoretical security, but it does not use any unproven
result from computational complexity. On the negative side, the protocol is not
efficient in the sense that the computation of the two legal parties uses more
space than the space allowed to the adversary.Comment: Several small errors have been fixed and the presentation has been
improved, following the reviewers' observation
Lifting query complexity to time-space complexity for two-way finite automata
Time-space tradeoff has been studied in a variety of models, such as Turing
machines, branching programs, and finite automata, etc. While communication
complexity as a technique has been applied to study finite automata, it seems
it has not been used to study time-space tradeoffs of finite automata. We
design a new technique showing that separations of query complexity can be
lifted, via communication complexity, to separations of time-space complexity
of two-way finite automata. As an application, one of our main results exhibits
the first example of a language such that the time-space complexity of
two-way probabilistic finite automata with a bounded error (2PFA) is
, while of exact two-way quantum finite automata with
classical states (2QCFA) is , that is, we demonstrate
for the first time that exact quantum computing has an advantage in time-space
complexity comparing to classical computing
Unbounded violation of tripartite Bell inequalities
We prove that there are tripartite quantum states (constructed from random
unitaries) that can lead to arbitrarily large violations of Bell inequalities
for dichotomic observables. As a consequence these states can withstand an
arbitrary amount of white noise before they admit a description within a local
hidden variable model. This is in sharp contrast with the bipartite case, where
all violations are bounded by Grothendieck's constant. We will discuss the
possibility of determining the Hilbert space dimension from the obtained
violation and comment on implications for communication complexity theory.
Moreover, we show that the violation obtained from generalized GHZ states is
always bounded so that, in contrast to many other contexts, GHZ states do in
this case not lead to extremal quantum correlations. The results are based on
tools from the theories of operator spaces and tensor norms which we exploit to
prove the existence of bounded but not completely bounded trilinear forms from
commutative C*-algebras.Comment: Substantial changes in the presentation to make the paper more
accessible for a non-specialized reade
The Sketching Complexity of Graph and Hypergraph Counting
Subgraph counting is a fundamental primitive in graph processing, with
applications in social network analysis (e.g., estimating the clustering
coefficient of a graph), database processing and other areas. The space
complexity of subgraph counting has been studied extensively in the literature,
but many natural settings are still not well understood. In this paper we
revisit the subgraph (and hypergraph) counting problem in the sketching model,
where the algorithm's state as it processes a stream of updates to the graph is
a linear function of the stream. This model has recently received a lot of
attention in the literature, and has become a standard model for solving
dynamic graph streaming problems.
In this paper we give a tight bound on the sketching complexity of counting
the number of occurrences of a small subgraph in a bounded degree graph
presented as a stream of edge updates. Specifically, we show that the space
complexity of the problem is governed by the fractional vertex cover number of
the graph . Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
. Our main technical contribution lies in a new set of Fourier analytic
tools that we develop to analyze multiplayer communication protocols in the
simultaneous communication model, allowing us to prove a tight lower bound. We
believe that our techniques are likely to find applications in other settings.
Besides giving tight bounds for all graphs , both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity
Communication Memento: Memoryless Communication Complexity
We study the communication complexity of computing functions
in the memoryless
communication model. Here, Alice is given , Bob is given and their goal is to compute F(x,y) subject to the following
constraint: at every round, Alice receives a message from Bob and her reply to
Bob solely depends on the message received and her input x; the same applies to
Bob. The cost of computing F in this model is the maximum number of bits
exchanged in any round between Alice and Bob (on the worst case input x,y). In
this paper, we also consider variants of our memoryless model wherein one party
is allowed to have memory, the parties are allowed to communicate quantum bits,
only one player is allowed to send messages. We show that our memoryless
communication model capture the garden-hose model of computation by Buhrman et
al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13)
and the overlay communication complexity by Papakonstantinou et al. (CCC'14).
Thus the memoryless communication complexity model provides a unified framework
to study space-bounded communication models. We establish the following: (1) We
show that the memoryless communication complexity of F equals the logarithm of
the size of the smallest bipartite branching program computing F (up to a
factor 2); (2) We show that memoryless communication complexity equals
garden-hose complexity; (3) We exhibit various exponential separations between
these memoryless communication models.
We end with an intriguing open question: can we find an explicit function F
and universal constant c>1 for which the memoryless communication complexity is
at least ? Note that would imply a
lower bound for general formula size, improving
upon the best lower bound by Ne\v{c}iporuk in 1966.Comment: 30 Pages; several improvements to the presentation
Streaming algorithms for language recognition problems
We study the complexity of the following problems in the streaming model.
Membership testing for \DLIN We show that every language in \DLIN\ can be
recognised by a randomized one-pass space algorithm with inverse
polynomial one-sided error, and by a deterministic p-pass space
algorithm. We show that these algorithms are optimal.
Membership testing for \LL For languages generated by \LL grammars
with a bound of on the number of nonterminals at any stage in the left-most
derivation, we show that membership can be tested by a randomized one-pass
space algorithm with inverse polynomial (in ) one-sided error.
Membership testing for \DCFL We show that randomized algorithms as efficient
as the ones described above for \DLIN\ and \LL(k) (which are subclasses of
\DCFL) cannot exist for all of \DCFL: there is a language in \VPL\ (a subclass
of \DCFL) for which any randomized p-pass algorithm with error bounded by
must use space.
Degree sequence problem We study the problem of determining, given a sequence
and a graph , whether the degree sequence of is
precisely . We give a randomized one-pass space
algorithm with inverse polynomial one-sided error probability. We show that our
algorithms are optimal.
Our randomized algorithms are based on the recent work of Magniez et al.
\cite{MMN09}; our lower bounds are obtained by considering related
communication complexity problems
Unitary Branching Programs: Learnability and Lower Bounds
Bounded width branching programs are a formalism that can be used to capture the notion of non-uniform constant-space computation. In this work, we study a generalized version of bounded width branching programs where instructions are defined by unitary matrices of bounded dimension. We introduce a new learning framework for these branching programs that leverages on a combination of local search techniques with gradient descent over Riemannian manifolds. We also show that gapped, read-once branching programs of bounded dimension can be learned with a polynomial number of queries in the presence of a teacher. Finally, we provide explicit near-quadratic size lower-bounds for bounded-dimension unitary branching programs, and exponential size lower-bounds for bounded-dimension read-once gapped unitary branching programs. The first lower bound is proven using a combination of Neciporuk’s lower bound technique with classic results from algebraic geometry. The second lower bound is proven within the framework of communication complexity theory.publishedVersio
Stochastic Streams: Sample Complexity vs. Space Complexity
We address the trade-off between the computational resources needed to process a large data set and the number of samples available from the data set. Specifically, we consider the following abstraction: we receive a potentially infinite stream of IID samples from some unknown distribution D, and are tasked with computing some function f(D). If the stream is observed for time t, how much memory, s, is required to estimate f(D)? We refer to t as the sample complexity and s as the space complexity. The main focus of this paper is investigating the trade-offs between the space and sample complexity. We study these trade-offs for several canonical problems studied in the data stream model: estimating the collision probability, i.e., the second moment of a distribution, deciding if a graph is connected, and approximating the dimension of an unknown subspace. Our results are based on techniques for simulating different classical sampling procedures in this model, emulating random walks given a sequence of IID samples, as well as leveraging a characterization between communication bounded protocols and statistical query algorithms
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