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

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 $c$

Optimal refrigerator

We study a refrigerator model which consists of two $n$-level systems interacting via a pulsed external field. Each system couples to its own thermal bath at temperatures $T_h$ and $T_c$, respectively ($\theta\equiv T_c/T_h<1$). The refrigerator functions in two steps: thermally isolated interaction between the systems driven by the external field and isothermal relaxation back to equilibrium. There is a complementarity between the power of heat transfer from the cold bath and the efficiency: the latter nullifies when the former is maximized and {\it vice versa}. A reasonable compromise is achieved by optimizing the product of the heat-power and efficiency over the Hamiltonian of the two system. The efficiency is then found to be bounded from below by $\zeta_{\rm CA}=\frac{1}{\sqrt{1-\theta}}-1$ (an analogue of the Curzon-Ahlborn efficiency), besides being bound from above by the Carnot efficiency $\zeta_{\rm C} = \frac{1}{1-\theta}-1$. The lower bound is reached in the equilibrium limit $\theta\to 1$. The Carnot bound is reached (for a finite power and a finite amount of heat transferred per cycle) for $\ln n\gg 1$. If the above maximization is constrained by assuming homogeneous energy spectra for both systems, the efficiency is bounded from above by $\zeta_{\rm CA}$ and converges to it for $n\gg 1$.Comment: 12 pages, 3 figure

Optimal Subharmonic Entrainment

For many natural and engineered systems, a central function or design goal is the synchronization of one or more rhythmic or oscillating processes to an external forcing signal, which may be periodic on a different time-scale from the actuated process. Such subharmonic synchrony, which is dynamically established when N control cycles occur for every M cycles of a forced oscillator, is referred to as N:M entrainment. In many applications, entrainment must be established in an optimal manner, for example by minimizing control energy or the transient time to phase locking. We present a theory for deriving inputs that establish subharmonic N:M entrainment of general nonlinear oscillators, or of collections of rhythmic dynamical units, while optimizing such objectives. Ordinary differential equation models of oscillating systems are reduced to phase variable representations, each of which consists of a natural frequency and phase response curve. Formal averaging and the calculus of variations are then applied to such reduced models in order to derive optimal subharmonic entrainment waveforms. The optimal entrainment of a canonical model for a spiking neuron is used to illustrate this approach, which is readily extended to arbitrary oscillating systems

Optimal mixing enhancement

We introduce a general-purpose method for optimising the mixing rate of advective fluid flows. An existing velocity field is perturbed in a $C^1$ neighborhood to maximize the mixing rate for flows generated by velocity fields in this neighborhood. Our numerical approach is based on the infinitesimal generator of the flow and is solved by standard linear programming methods. The perturbed flow may be easily constrained to preserve the same steady state distribution as the original flow, and various natural geometric constraints can also be simply applied. The same technique can also be used to optimize the mixing rate of advection-diffusion flow models by manipulating the drift term in a small neighborhood

Optimal market making

Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. They face a complex optimization problem with both static and dynamic components. They need indeed to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread). Since they seldom buy and sell simultaneously, and therefore hold long and/or short inventories, they also need to mitigate the risk associated with price changes, and subsequently skew their quotes dynamically. In this paper, (i) we propose a general modeling framework which generalizes (and reconciles) the various modeling approaches proposed in the literature since the publication of the seminal paper "High-frequency trading in a limit order book" by Avellaneda and Stoikov, (ii) we prove new general results on the existence and the characterization of optimal market making strategies, (iii) we obtain new closed-form approximations for the optimal quotes, (iv) we extend the modeling framework to the case of multi-asset market making and we obtain general closed-form approximations for the optimal quotes of a multi-asset market maker, and (v) we show how the model can be used in practice in the specific (and original) case of two credit indices

Optimal reduction

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit reduction procedure for canonical actions on a Poisson manifold that exhibits an interesting interplay with the von Neumann condition previously introduced by the author in his study of singular dual pairs. This condition ensures that the orbits in the momentum space of the optimal momentum map (we call them polar reduced spaces) admit a presymplectic structure that generalizes the Kostant--Kirillov--Souriau symplectic structure of the coadjoint orbits in the dual of a Lie algebra. Using this presymplectic structure, the optimal orbit reduced spaces are symplectic with a form that satisfies a relation identical to the classical one obtained by Marle, Kazhdan, Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold. In the symplectic case we provide a necessary and sufficient condition for the polar reduced spaces to be symplectic. In general, the presymplectic polar reduced spaces are foliated by symplectic submanifolds that are obtained through a generalization to the optimal context of the so called Sjamaar Principle, already existing in the theory of Hamiltonian singular reduction. We use these ideas in the construction of a family of presymplectic homogeneous manifolds and of its symplectic foliation and we show that these reduction techniques can be implemented in stages in total analogy with the case of free globally Hamiltonian proper actions.Comment: 42 page

Optimal states and almost optimal adaptive measurements for quantum interferometry

We derive the optimal N-photon two-mode input state for obtaining an estimate \phi of the phase difference between two arms of an interferometer. For an optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995)], it yields a variance (\Delta \phi)^2 \simeq \pi^2/N^2, compared to O(N^{-1}) or O(N^{-1/2}) for states considered by previous authors. Such a measurement cannot be realized by counting photons in the interferometer outputs. However, we introduce an adaptive measurement scheme that can be thus realized, and show that it yields a variance in \phi very close to that from an optimal measurement.Comment: 4 pages, 4 figures, journal versio