117 research outputs found
Lower Bounds on the Communication Complexity of Binary Local Quantum Measurement Simulation
We consider the problem of the classical simulation of quantum measurements
in the scenario of communication complexity. Regev and Toner (2007) have
presented a 2-bit protocol which simulates one particular correlation function
arising from binary projective quantum measurements on arbitrary state, and in
particular does not preserve local averages. The question of simulating other
correlation functions using a protocol with bounded communication, or
preserving local averages, has been posed as an open one. Within this paper we
resolve it in the negative: we show that any such protocol must have unbounded
communication for some subset of executions. In particular, we show that for
any protocol, there exist inputs for which the random variable describing the
number of communicated bits has arbitrarily large variance
Distinguishing Views in Symmetric Networks: A Tight Lower Bound
The view of a node in a port-labeled network is an infinite tree encoding all
walks in the network originating from this node. We prove that for any integers
, there exists a port-labeled network with at most nodes and
diameter at most which contains a pair of nodes whose (infinite) views are
different, but whose views truncated to depth are
identical
Local Conflict Coloring
Locally finding a solution to symmetry-breaking tasks such as
vertex-coloring, edge-coloring, maximal matching, maximal independent set,
etc., is a long-standing challenge in distributed network computing. More
recently, it has also become a challenge in the framework of centralized local
computation. We introduce conflict coloring as a general symmetry-breaking task
that includes all the aforementioned tasks as specific instantiations ---
conflict coloring includes all locally checkable labeling tasks from
[Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two
parameters and , where the former measures the amount of freedom given
to the nodes for selecting their colors, and the latter measures the number of
constraints which colors of adjacent nodes are subject to.We show that, in the
standard LOCAL model for distributed network computing, if l/d \textgreater{}
\Delta, then conflict coloring can be solved in rounds in -node graphs with maximum degree
, where ignores the polylog factors in . The
dependency in~ is optimal, as a consequence of the lower
bound by [Linial, SIAM J. Comp. 1992] for -coloring. An important
special case of our result is a significant improvement over the best known
algorithm for distributed -coloring due to [Barenboim, PODC 2015],
which required rounds. Improvements for other
variants of coloring, including -list-coloring,
-edge-coloring, -coloring, etc., also follow from our general
result on conflict coloring. Likewise, in the framework of centralized local
computation algorithms (LCAs), our general result yields an LCA which requires
a smaller number of probes than the previously best known algorithm for
vertex-coloring, and works for a wide range of coloring problems
Non-classicality of temporal correlations
The results of space-like separated measurements are independent of distant
measurement settings, a property one might call two-way no-signalling. In
contrast, time-like separated measurements are only one-way no-signalling since
the past is independent of the future but not vice-versa. For this reason
temporal correlations that are formally identical to non-classical spatial
correlations can still be modelled classically. We define non-classical
temporal correlations as the ones which cannot be simulated by propagating in
time a classical information content of a quantum system. We first show that
temporal correlations between results of any projective quantum measurements on
a qubit can be simulated classically. Then we present a sequence of POVM
measurements on a single -level quantum system that cannot be explained by
propagating in time -level classical system and using classical computers
with unlimited memory.Comment: 6 pages, 1 figur
Interconnection network with a shared whiteboard: Impact of (a)synchronicity on computing power
In this work we study the computational power of graph-based models of
distributed computing in which each node additionally has access to a global
whiteboard. A node can read the contents of the whiteboard and, when activated,
can write one message of O(log n) bits on it. When the protocol terminates,
each node computes the output based on the final contents of the whiteboard. We
consider several scheduling schemes for nodes, providing a strict ordering of
their power in terms of the problems which can be solved with exactly one
activation per node. The problems used to separate the models are related to
Maximal Independent Set, detection of cycles of length 4, and BFS spanning tree
constructions
A Point Set Connection Problem for Autonomous Mobile Robots in a Grid
Consider an orthogonal grid of streets and avenues in a Manhattan-like city populated by stationary sensor modules at some intersections and mobile robots that can serve as relays of information that the modules exchange, where both module-module and module-robot communication is limited to a straight line of sight within the grid. The robots are oblivious and move asynchronously. We present a distributed algorithm that, given the sensor locations as input, moves the robots to suitable locations in the grid so that a connected network of all modules is established. The number of robots that the algorithm uses is worst case optimal
Improved Analysis of Deterministic Load-Balancing Schemes
We consider the problem of deterministic load balancing of tokens in the
discrete model. A set of processors is connected into a -regular
undirected network. In every time step, each processor exchanges some of its
tokens with each of its neighbors in the network. The goal is to minimize the
discrepancy between the number of tokens on the most-loaded and the
least-loaded processor as quickly as possible.
Rabani et al. (1998) present a general technique for the analysis of a wide
class of discrete load balancing algorithms. Their approach is to characterize
the deviation between the actual loads of a discrete balancing algorithm with
the distribution generated by a related Markov chain. The Markov chain can also
be regarded as the underlying model of a continuous diffusion algorithm. Rabani
et al. showed that after time , any algorithm of their
class achieves a discrepancy of , where is the spectral
gap of the transition matrix of the graph, and is the initial load
discrepancy in the system.
In this work we identify some natural additional conditions on deterministic
balancing algorithms, resulting in a class of algorithms reaching a smaller
discrepancy. This class contains well-known algorithms, eg., the Rotor-Router.
Specifically, we introduce the notion of cumulatively fair load-balancing
algorithms where in any interval of consecutive time steps, the total number of
tokens sent out over an edge by a node is the same (up to constants) for all
adjacent edges. We prove that algorithms which are cumulatively fair and where
every node retains a sufficient part of its load in each step, achieve a
discrepancy of in time . We
also show that in general neither of these assumptions may be omitted without
increasing discrepancy. We then show by a combinatorial potential reduction
argument that any cumulatively fair scheme satisfying some additional
assumptions achieves a discrepancy of almost as quickly as the
continuous diffusion process. This positive result applies to some of the
simplest and most natural discrete load balancing schemes.Comment: minor corrections; updated literature overvie
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