345 research outputs found
Improved Time-Space Trade-Offs for Computing Voronoi Diagrams
Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space.
For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards.
We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques
Improved time-space trade-offs for computing Voronoi diagrams
Let P
be a planar set of n sites in general position. For kâ{1,âŠ,nâ1}, the Voronoi diagram of order k for P is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest k neighbors in P. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of k=1 and k=nâ1, respectively. For any given Kâ{1,âŠ,nâ1}, the family of all higher-order Voronoi diagrams of order k=1,âŠ,K for P can be computed in total time O(nK2+nlogn) using O(K2(nâK)) space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for P can be computed in O(nlogn) time using O(n)
space [Preparata, Shamos, Springer'85].
For sâ{1,âŠ,n}
, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words, of Î(logn)
bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards.
We describe a deterministic s
-workspace algorithm for computing NVD and FVD for P that runs in O((n2/s)logs) time. Moreover, we generalize our s-workspace algorithm so that for any given KâO(sâ), we compute the family of all higher-order Voronoi diagrams of order k=1,âŠ,K for P in total expected time O(n2K5s(logs+K2O(logâK))) or in total deterministic time O(n2K5s(logs+KlogK)). Previously, for Voronoi diagrams, the only known s-workspace algorithm runs in expected time O((n2/s)logs+nlogslogâs) [Korman et al., WADS'15] and only works for NVD (i.e., k=1). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques
Routing in Polygonal Domains
We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex. Then, we must be able to route a data packet between any two vertices p and q of Pwhere each step must use only the label of the target node q and the routing table of the current node.
For any fixed eps > 0, we pre ent a routing scheme that always achieves a routing path that exceeds the shortest path by a factor of at most 1 + eps. The labels have O(log n) bits, and the routing tables are of size O((eps^{-1} + h) log n). The preprocessing time is O(n^2 log n + hn^2 + eps^{-1}hn). It can be improved to O(n 2 + eps^{-1}n) for simple polygons
Investigation of two Fermi-LAT gamma-ray blazars coincident with high-energy neutrinos detected by IceCube
After the identification of the gamma-ray blazar TXS 0506+056 as the first
compelling IceCube neutrino source candidate, we perform a systematic analysis
of all high-energy neutrino events satisfying the IceCube realtime trigger
criteria. We find one additional known gamma-ray source, the blazar GB6
J1040+0617, in spatial coincidence with a neutrino in this sample. The chance
probability of this coincidence is 30% after trial correction. For the first
time, we present a systematic study of the gamma-ray flux, spectral and optical
variability, and multi-wavelength behavior of GB6 J1040+0617 and compare it to
TXS 0506+056. We find that TXS 0506+056 shows strong flux variability in the
Fermi-LAT gamma-ray band, being in an active state around the arrival of
IceCube-170922A, but in a low state during the archival IceCube neutrino flare
in 2014/15. In both cases the spectral shape is statistically compatible () with the average spectrum showing no indication of a significant
relative increase of a high-energy component. While the association of GB6
J1040+0617 with the neutrino is consistent with background expectations, the
source appears to be a plausible neutrino source candidate based on its
energetics and multi-wavelength features, namely a bright optical flare and
modestly increased gamma-ray activity. Finding one or two neutrinos originating
from gamma-ray blazars in the given sample of high-energy neutrinos is
consistent with previously derived limits of neutrino emission from gamma-ray
blazars, indicating the sources of the majority of cosmic high-energy neutrinos
remain unknown.Comment: 22 pages, 11 figures, 2 Table
LeptonInjector and LeptonWeighter: A neutrino event generator and weighter for neutrino observatories
We present a high-energy neutrino event generator, called LeptonInjector,
alongside an event weighter, called LeptonWeighter. Both are designed for
large-volume Cherenkov neutrino telescopes such as IceCube. The neutrino event
generator allows for quick and flexible simulation of neutrino events within
and around the detector volume, and implements the leading Standard Model
neutrino interaction processes relevant for neutrino observatories:
neutrino-nucleon deep-inelastic scattering and neutrino-electron annihilation.
In this paper, we discuss the event generation algorithm, the weighting
algorithm, and the main functions of the publicly available code, with
examples.Comment: 28 pages, 10 figures, 3 table
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