4,481,208 research outputs found
Optimal solution of nonlinear equations
Journal ArticleWe survey recent worst case complexity results for the solution of nonlinear equations. Notes on worst and average case analysis of iterative algorithms and a bibliography of the subject are also included
Quantum Weak Coin Flipping
We investigate weak coin flipping, a fundamental cryptographic primitive
where two distrustful parties need to remotely establish a shared random bit. A
cheating player can try to bias the output bit towards a preferred value. For
weak coin flipping the players have known opposite preferred values. A weak
coin-flipping protocol has a bias if neither player can force the
outcome towards their preferred value with probability more than
. While it is known that all classical protocols have
, Mochon showed in 2007 [arXiv:0711.4114] that quantumly
weak coin flipping can be achieved with arbitrarily small bias (near perfect)
but the best known explicit protocol has bias (also due to Mochon, 2005
[Phys. Rev. A 72, 022341]). We propose a framework to construct new explicit
protocols achieving biases below . In particular, we construct explicit
unitaries for protocols with bias approaching . To go below, we introduce
what we call the Elliptic Monotone Align (EMA) algorithm which, together with
the framework, allows us to numerically construct protocols with arbitrarily
small biases.Comment: 98 pages split into 3 parts, 10 figures; For updates and contact
information see https://atulsingharora.github.io/WCF. Version 2 has minor
improvements. arXiv admin note: text overlap with arXiv:1402.7166 by other
author
String Synchronizing Sets: Sublinear-Time BWT Construction and Optimal LCE Data Structure
Burrows-Wheeler transform (BWT) is an invertible text transformation that,
given a text of length , permutes its symbols according to the
lexicographic order of suffixes of . BWT is one of the most heavily studied
algorithms in data compression with numerous applications in indexing, sequence
analysis, and bioinformatics. Its construction is a bottleneck in many
scenarios, and settling the complexity of this task is one of the most
important unsolved problems in sequence analysis that has remained open for 25
years. Given a binary string of length , occupying machine
words, the BWT construction algorithm due to Hon et al. (SIAM J. Comput., 2009)
runs in time and space. Recent advancements (Belazzougui,
STOC 2014, and Munro et al., SODA 2017) focus on removing the alphabet-size
dependency in the time complexity, but they still require time.
In this paper, we propose the first algorithm that breaks the -time
barrier for BWT construction. Given a binary string of length , our
procedure builds the Burrows-Wheeler transform in time and
space. We complement this result with a conditional lower bound
proving that any further progress in the time complexity of BWT construction
would yield faster algorithms for the very well studied problem of counting
inversions: it would improve the state-of-the-art -time
solution by Chan and P\v{a}tra\c{s}cu (SODA 2010). Our algorithm is based on a
novel concept of string synchronizing sets, which is of independent interest.
As one of the applications, we show that this technique lets us design a data
structure of the optimal size that answers Longest Common
Extension queries (LCE queries) in time and, furthermore, can be
deterministically constructed in the optimal time.Comment: Full version of a paper accepted to STOC 201
Replica Placement on Bounded Treewidth Graphs
We consider the replica placement problem: given a graph with clients and
nodes, place replicas on a minimum set of nodes to serve all the clients; each
client is associated with a request and maximum distance that it can travel to
get served and there is a maximum limit (capacity) on the amount of request a
replica can serve. The problem falls under the general framework of capacitated
set covering. It admits an O(\log n)-approximation and it is NP-hard to
approximate within a factor of . We study the problem in terms of
the treewidth of the graph and present an O(t)-approximation algorithm.Comment: An abridged version of this paper is to appear in the proceedings of
WADS'1
Approximate Near Neighbors for General Symmetric Norms
We show that every symmetric normed space admits an efficient nearest
neighbor search data structure with doubly-logarithmic approximation.
Specifically, for every , , and every -dimensional
symmetric norm , there exists a data structure for
-approximate nearest neighbor search over
for -point datasets achieving query time and
space. The main technical ingredient of the algorithm is a
low-distortion embedding of a symmetric norm into a low-dimensional iterated
product of top- norms.
We also show that our techniques cannot be extended to general norms.Comment: 27 pages, 1 figur
Optimal pricing for optimal transport
Suppose that is the cost of transporting a unit of mass from to and suppose that a mass distribution on is transported
optimally (so that the total cost of transportation is minimal) to the mass
distribution on . Then, roughly speaking, the Kantorovich duality
theorem asserts that there is a price for a unit of mass sold (say by
the producer to the distributor) at and a price for a unit of mass
sold (say by the distributor to the end consumer) at such that for any
and , the price difference is not greater than the
cost of transportation and such that there is equality
if indeed a nonzero mass was transported (via the optimal
transportation plan) from to . We consider the following optimal pricing
problem: suppose that a new pricing policy is to be determined while keeping a
part of the optimal transportation plan fixed and, in addition, some prices at
the sources of this part are also kept fixed. From the producers' side, what
would then be the highest compatible pricing policy possible? From the
consumers' side, what would then be the lowest compatible pricing policy
possible? In the framework of -convexity theory, we have recently introduced
and studied optimal -convex -antiderivatives and explicit constructions
of these optimizers were presented. In the present paper we employ optimal
-convex -antiderivatives and conclude that these are natural solutions to
the optimal pricing problems mentioned above. This type of problems drew
attention in the past and existence results were previously established in the
case where under various specifications. We solve the above problem
for general spaces and real-valued, lower semicontinuous cost functions
From optimal transportation to optimal teleportation
The object of this paper is to study estimates of
for small . Here is
the Wasserstein metric on positive measures, , is a probability
measure and a signed, neutral measure (). In [W1] we proved
uniform (in ) estimates for provided can be
controlled in terms of the , for any smooth
function .
In this paper we extend the results to the case where such a control fails.
This is the case where if, e.g. has a disconnected support, or if the
dimension of , (to be defined) is larger or equal .
In the later case we get such an estimate provided for
. If we get a log-Lipschitz estimate.
As an application we obtain H\"{o}lder estimates in for curves of
probability measures which are absolutely continuous in the total variation
norm .
In case the support of is disconnected (corresponding to ) we
obtain sharp estimates for ("optimal teleportation"): where is expressed in terms of optimal
transport on a metric graph, determined only by the relative distances between
the connected components of the support of , and the weights of the
measure in each connected component of this support.Comment: 24 pages, 3 figure
Nonlinear convex and concave relaxations for the solutions of parametric ODEs
SUMMARY Convex and concave relaxations for the parametric solutions of ordinary differential equations (ODEs) are central to deterministic global optimization methods for nonconvex dynamic optimization and open-loop optimal control problems with control parametrization. Given a general system of ODEs with parameter dependence in the initial conditions and right-hand sides, this work derives sufficient conditions under which an auxiliary system of ODEs describes convex and concave relaxations of the parametric solutions, pointwise in the independent variable. Convergence results for these relaxations are also established. A fully automatable procedure for constructing an appropriate auxiliary system has been developed previously by the authors. Thus, the developments here lead to an efficient, automatic method for computing convex and concave relaxations for the parametric solutions of a very general class nonlinear ODEs. The proposed method is presented in detail for a simple example problem
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