1,621 research outputs found
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 , we show how to
construct in time a data structure of size that answers
distance queries in time per query.
As a consequence, we obtain an improvement over the fastest algorithm for
k-many distances in planar graphs whenever .
* We provide a linear-space exact distance oracle for planar graphs with
query time 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
such that for any pair of nodes at distance D the query time is
. 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 nodes, we can preprocess G in
time to produce a data structure of size that can
answer the following queries in 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
and space , we essentially improve the query
time from to .Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on
Discrete Algorithms, SODA 201
Exploiting -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 , the ratio of the square of the
coherent cavity coupling strength over the product of cavity and emitter decay
rates. For a single emitter, 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
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