574 research outputs found
On the parity complexity measures of Boolean functions
The parity decision tree model extends the decision tree model by allowing
the computation of a parity function in one step. We prove that the
deterministic parity decision tree complexity of any Boolean function is
polynomially related to the non-deterministic complexity of the function or its
complement. We also show that they are polynomially related to an analogue of
the block sensitivity. We further study parity decision trees in their
relations with an intermediate variant of the decision trees, as well as with
communication complexity.Comment: submitted to TCS on 16-MAR-200
On Communication Protocols that Compute Almost Privately
A traditionally desired goal when designing auction mechanisms is incentive
compatibility, i.e., ensuring that bidders fare best by truthfully reporting
their preferences. A complementary goal, which has, thus far, received
significantly less attention, is to preserve privacy, i.e., to ensure that
bidders reveal no more information than necessary. We further investigate and
generalize the approximate privacy model for two-party communication recently
introduced by Feigenbaum et al.[8]. We explore the privacy properties of a
natural class of communication protocols that we refer to as "dissection
protocols". Dissection protocols include, among others, the bisection auction
in [9,10] and the bisection protocol for the millionaires problem in [8].
Informally, in a dissection protocol the communicating parties are restricted
to answering simple questions of the form "Is your input between the values
\alpha and \beta (under a predefined order over the possible inputs)?".
We prove that for a large class of functions, called tiling functions, which
include the 2nd-price Vickrey auction, there always exists a dissection
protocol that provides a constant average-case privacy approximation ratio for
uniform or "almost uniform" probability distributions over inputs. To establish
this result we present an interesting connection between the approximate
privacy framework and basic concepts in computational geometry. We show that
such a good privacy approximation ratio for tiling functions does not, in
general, exist in the worst case. We also discuss extensions of the basic setup
to more than two parties and to non-tiling functions, and provide calculations
of privacy approximation ratios for two functions of interest.Comment: to appear in Theoretical Computer Science (series A
Two-Party Direct-Sum Questions Through the Lens of Multiparty Communication Complexity
Direct-sum questions in (two-party) communication complexity ask whether two parties, Alice and Bob, can compute the value of a function f on l inputs (x_1,y_1),...,(x_l,y_l) more efficiently than by applying the best protocol for f, independently on each input (x_i,y_i). In spite of significant efforts to understand these questions (under various communication-complexity measures), the general question is still far from being well understood.
In this paper, we offer a multiparty view of these questions: The direct-sum setting is just a two-player system with Alice having inputs x_1,...,x_l, Bob having inputs y_1,...,y_l and the desired output is f(x_1,y_1),...,f(x_l,y_l). The naive solution of solving the l problems independently, is modeled by a network with l (disconnected) pairs of players Alice i and Bob i, with inputs x_i,y_i respectively, and communication only within each pair. Then, we consider an intermediate ("star") model, where there is one Alice having l inputs x_1,...,x_l and l players Bob_1,...,Bob_l holding y_1,...,y_l, respectively (in fact, we consider few variants of this intermediate model, depending on whether communication between each Bob i and Alice is point-to-point or whether we allow broadcast). Our goal is to get a better understanding of the relation between the two extreme models (i.e., of the two-party direct-sum question). If, for instance, Alice and Bob can do better (for some complexity measure) than solving the l problems independently, we wish to understand what intermediate model already allows to do so (hereby understanding the "source" of such savings). If, on the other hand, we wish to prove that there is no better solution than solving the l problems independently, then our approach gives a way of breaking the task of proving such a statement into few (hopefully, easier) steps.
We present several results of both types. Namely, for certain complexity measures, communication problems f and certain pairs of models, we can show gaps between the complexity of solving f on l instances in the two models in question; while, for certain other complexity measures and pairs of models, we can show that such gaps do not exist (for any communication problem f). For example, we prove that if only point-to-point communication is allowed in the intermediate "star" model, then significant savings are impossible in the public-coin randomized setting. On the other hand, in the private-coin randomized setting, if Alice is allowed to broadcast messages to all Bobs in the "star" network, then some savings are possible. While this approach does not lead yet to new results on the original two-party direct-sum question, we believe that our work gives new insights on the already-known direct-sum results, and may potentially lead to more such results in the future
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
Information complexity of the AND function in the two-Party, and multiparty settings
In a recent breakthrough paper [M. Braverman, A. Garg, D. Pankratov, and O.
Weinstein, From information to exact communication, STOC'13] Braverman et al.
developed a local characterization for the zero-error information complexity in
the two party model, and used it to compute the exact internal and external
information complexity of the 2-bit AND function, which was then applied to
determine the exact asymptotic of randomized communication complexity of the
set disjointness problem.
In this article, we extend their results on AND function to the multi-party
number-in-hand model by proving that the generalization of their protocol has
optimal internal and external information cost for certain distributions. Our
proof has new components, and in particular it fixes some minor gaps in the
proof of Braverman et al
New bounds on classical and quantum one-way communication complexity
In this paper we provide new bounds on classical and quantum distributional
communication complexity in the two-party, one-way model of communication. In
the classical model, our bound extends the well known upper bound of Kremer,
Nisan and Ron to include non-product distributions. We show that for a boolean
function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and
epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where
D_{epsilon}^{1, mu}(f) represents the one-way distributional communication
complexity of f with error at most epsilon under mu; vc(f) represents the
Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual
information, under mu, between the random inputs of the two parties. For a
non-boolean function f:X x Y ->[k], we show a similar upper bound on
D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f' =
f/k. In the quantum one-way model we provide a lower bound on the
distributional communication complexity, under product distributions, of a
function f, in terms the well studied complexity measure of f referred to as
the rectangle bound or the corruption bound of f . We show for a non-boolean
total function f : X x Y -> Z and a product distribution mu on XxY,
Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where
Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional
communication complexity of f with error at most epsilon^3/8 under mu and rec_
epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at
most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z
U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x
15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).Comment: ver 1, 19 page
Proposed experiment for the quantum "Guess my number" protocol
An experimental realization of the entanglement-assisted "Guess my number"
protocol for the reduction of communication complexity, introduced by Steane
and van Dam, would require producing and detecting three-qubit GHZ states with
an efficiency eta > 0.70, which would require single photon detectors of
efficiency sigma > 0.89. We propose a modification of the protocol which can be
translated into a real experiment using present-day technology. In the proposed
experiment, the quantum reduction of the multi-party communication complexity
would require an efficiency eta > 0.05, achievable with detectors of sigma >
0.47, for four parties, and eta > 0.17 (sigma > 0.55) for three parties.Comment: REVTeX4, 4 pages, 1 figur
The Communication Complexity of the Hamming Distance Problem
We investigate the randomized and quantum communication complexity of the
Hamming Distance problem, which is to determine if the Hamming distance between
two n-bit strings is no less than a threshold d. We prove a quantum lower bound
of \Omega(d) qubits in the general interactive model with shared prior
entanglement. We also construct a classical protocol of O(d \log d) bits in the
restricted Simultaneous Message Passing model, improving previous protocols of
O(d^2) bits (A. C.-C. Yao, Proceedings of the Thirty-Fifth Annual ACM Symposium
on Theory of Computing, pp. 77-81, 2003), and O(d\log n) bits (D. Gavinsky, J.
Kempe, and R. de Wolf, quant-ph/0411051, 2004).Comment: 8 pages, v3, updated reference. to appear in Information Processing
Letters, 200
Distributed Deterministic Broadcasting in Uniform-Power Ad Hoc Wireless Networks
Development of many futuristic technologies, such as MANET, VANET, iThings,
nano-devices, depend on efficient distributed communication protocols in
multi-hop ad hoc networks. A vast majority of research in this area focus on
design heuristic protocols, and analyze their performance by simulations on
networks generated randomly or obtained in practical measurements of some
(usually small-size) wireless networks. %some library. Moreover, they often
assume access to truly random sources, which is often not reasonable in case of
wireless devices. In this work we use a formal framework to study the problem
of broadcasting and its time complexity in any two dimensional Euclidean
wireless network with uniform transmission powers. For the analysis, we
consider two popular models of ad hoc networks based on the
Signal-to-Interference-and-Noise Ratio (SINR): one with opportunistic links,
and the other with randomly disturbed SINR. In the former model, we show that
one of our algorithms accomplishes broadcasting in rounds, where
is the number of nodes and is the diameter of the network. If nodes
know a priori the granularity of the network, i.e., the inverse of the
maximum transmission range over the minimum distance between any two stations,
a modification of this algorithm accomplishes broadcasting in
rounds.
Finally, we modify both algorithms to make them efficient in the latter model
with randomly disturbed SINR, with only logarithmic growth of performance.
Ours are the first provably efficient and well-scalable, under the two
models, distributed deterministic solutions for the broadcast task.Comment: arXiv admin note: substantial text overlap with arXiv:1207.673
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