Exact Distance Oracles for Planar Graphs

We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation $S\in[n\lg\lg n,n^2]$, we show how to construct in $\tilde O(S)$ time a data structure of size $O(S)$ that answers distance queries in $\tilde O(n/\sqrt S)$ time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever $k\in[\sqrt n,n)$. * We provide a linear-space exact distance oracle for planar graphs with query time $O(n^{1/2+eps})$ for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space $\tilde O(n)$ such that for any pair of nodes at distance D the query time is $\tilde O(min {D,\sqrt n})$. Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with $c = O(\sqrt n)$ nodes, we can preprocess G in $\tilde O(n)$ time to produce a data structure of size $O(n \lg\lg c)$ that can answer the following queries in $\tilde O(c)$ time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA'05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For $\sigma\in(1,4/3)$ and space $S=n^\sigma$, we essentially improve the query time from $n^2/S$ to $\sqrt{n^2/S}$.Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, SODA 201

Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

Prospects of reinforcement learning for the simultaneous damping of many mechanical modes

We apply adaptive feedback for the partial refrigeration of a mechanical resonator, i.e. with the aim to simultaneously cool the classical thermal motion of more than one vibrational degree of freedom. The feedback is obtained from a neural network parametrized policy trained via a reinforcement learning strategy to choose the correct sequence of actions from a finite set in order to simultaneously reduce the energy of many modes of vibration. The actions are realized either as optical modulations of the spring constants in the so-called quadratic optomechanical coupling regime or as radiation pressure induced momentum kicks in the linear coupling regime. As a proof of principle we numerically illustrate efficient simultaneous cooling of four independent modes with an overall strong reduction of the total system temperature.Comment: Machine learning in Optomechanics: coolin

All-Pairs Approximate Shortest Paths and Distance Oracle Preprocessing

Given an undirected, unweighted graph G on n nodes, there is an O(n^2*poly log(n))-time algorithm that computes a data structure called distance oracle of size O(n^{5/3}*poly log(n)) answering approximate distance queries in constant time. For nodes at distance d the distance estimate is between d and 2d + 1. This new distance oracle improves upon the oracles of Patrascu and Roditty (FOCS 2010), Abraham and Gavoille (DISC 2011), and Agarwal and Brighten Godfrey (PODC 2013) in terms of preprocessing time, and upon the oracle of Baswana and Sen (SODA 2004) in terms of stretch. The running time analysis is tight (up to logarithmic factors) due to a recent lower bound of Abboud and Bodwin (STOC 2016). Techniques include dominating sets, sampling, balls, and spanners, and the main contribution lies in the way these techniques are combined. Perhaps the most interesting aspect from a technical point of view is the application of a spanner without incurring its constant additive stretch penalty

Enhanced collective Purcell effect of coupled quantum emitter systems

Cavity-embedded quantum emitters show strong modifications of free space radiation properties such as an enhanced decay known as the Purcell effect. The central parameter is the cooperativity $C$, the ratio of the square of the coherent cavity coupling strength over the product of cavity and emitter decay rates. For a single emitter, $C$ is independent of the transition dipole moment and dictated by geometric cavity properties such as finesse and mode waist. In a recent work [Phys. Rev. Lett. 119, 093601 (2017)] we have shown that collective excitations in ensembles of dipole-dipole coupled quantum emitters show a disentanglement between the coherent coupling to the cavity mode and spontaneous free space decay. This leads to a strong enhancement of the cavity cooperativity around certain collective subradiant antiresonances. Here, we present a quantum Langevin equations approach aimed at providing results beyond the classical coupled dipoles model. We show that the subradiantly enhanced cooperativity imprints its effects onto the cavity output field quantum correlations while also strongly increasing the cavity-emitter system's collective Kerr nonlinear effect