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 $\epsilon$ if neither player can force the outcome towards their preferred value with probability more than $\frac{1}{2}+\epsilon$. While it is known that all classical protocols have $\epsilon=\frac{1}{2}$, 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 $1/6$ (also due to Mochon, 2005 [Phys. Rev. A 72, 022341]). We propose a framework to construct new explicit protocols achieving biases below $1/6$. In particular, we construct explicit unitaries for protocols with bias approaching $1/10$. 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 $T$ of length $n$, permutes its symbols according to the lexicographic order of suffixes of $T$. 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 $n$, occupying $O(n/\log n)$ machine words, the BWT construction algorithm due to Hon et al. (SIAM J. Comput., 2009) runs in $O(n)$ time and $O(n/\log n)$ 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 $\Omega(n)$ time. In this paper, we propose the first algorithm that breaks the $O(n)$-time barrier for BWT construction. Given a binary string of length $n$, our procedure builds the Burrows-Wheeler transform in $O(n/\sqrt{\log n})$ time and $O(n/\log n)$ 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 $O(m\sqrt{\log m})$-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 $O(n/\log n)$ that answers Longest Common Extension queries (LCE queries) in $O(1)$ time and, furthermore, can be deterministically constructed in the optimal $O(n/\log n)$ 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 $o(\log n)$. We study the problem in terms of the treewidth $t$ 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 $n$, $d = n^{o(1)}$, and every $d$-dimensional symmetric norm $\|\cdot\|$, there exists a data structure for $\mathrm{poly}(\log \log n)$-approximate nearest neighbor search over $\|\cdot\|$ for $n$-point datasets achieving $n^{o(1)}$ query time and $n^{1+o(1)}$ 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-$k$ 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 $c(x,y)$ is the cost of transporting a unit of mass from $x\in X$ to $y\in Y$ and suppose that a mass distribution $\mu$ on $X$ is transported optimally (so that the total cost of transportation is minimal) to the mass distribution $\nu$ on $Y$. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price $f(x)$ for a unit of mass sold (say by the producer to the distributor) at $x$ and a price $g(y)$ for a unit of mass sold (say by the distributor to the end consumer) at $y$ such that for any $x\in X$ and $y\in Y$, the price difference $g(y)-f(x)$ is not greater than the cost of transportation $c(x,y)$ and such that there is equality $g(y)-f(x)=c(x,y)$ if indeed a nonzero mass was transported (via the optimal transportation plan) from $x$ to $y$. 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 $c$-convexity theory, we have recently introduced and studied optimal $c$-convex $c$-antiderivatives and explicit constructions of these optimizers were presented. In the present paper we employ optimal $c$-convex $c$-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 $X=Y=R^n$ under various specifications. We solve the above problem for general spaces $X,Y$ and real-valued, lower semicontinuous cost functions $c$

From optimal transportation to optimal teleportation

The object of this paper is to study estimates of $\epsilon^{-q}W_p(\mu+\epsilon\nu, \mu)$ for small $\epsilon>0$. Here $W_p$ is the Wasserstein metric on positive measures, $p>1$, $\mu$ is a probability measure and $\nu$ a signed, neutral measure ($\int d\nu=0$). In [W1] we proved uniform (in $\epsilon$) estimates for $q=1$ provided $\int \phi d\nu$ can be controlled in terms of the $\int|\nabla\phi|^{p/(p-1)}d\mu$, for any smooth function $\phi$. In this paper we extend the results to the case where such a control fails. This is the case where if, e.g. $\mu$ has a disconnected support, or if the dimension of $\mu$ , $d$ (to be defined) is larger or equal $p/(p-1)$. In the later case we get such an estimate provided $1/p+1/d\not=1$ for $q=\min(1, 1/p+1/d)$. If $1/p+1/d=1$ we get a log-Lipschitz estimate. As an application we obtain H\"{o}lder estimates in $W_p$ for curves of probability measures which are absolutely continuous in the total variation norm . In case the support of $\mu$ is disconnected (corresponding to $d=\infty$) we obtain sharp estimates for $q=1/p$ ("optimal teleportation"): $$\lim_{\epsilon\rightarrow 0}\epsilon^{-1/p}W_p(\mu, \mu+\epsilon\nu) = \|\nu\|_{\mu}$$ where $\|\nu\|_{\mu}$ 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 $\mu$, and the weights of the measure $\nu$ 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