6,421 research outputs found
Exact Distance Oracles for Planar Graphs with Failing Vertices
We consider exact distance oracles for directed weighted planar graphs in the
presence of failing vertices. Given a source vertex , a target vertex
and a set of failed vertices, such an oracle returns the length of a
shortest -to- path that avoids all vertices in . We propose oracles
that can handle any number of failures. More specifically, for a directed
weighted planar graph with vertices, any constant , and for any , we propose an oracle of size
that answers queries in
time. In particular, we show an
-size, -query-time
oracle for any constant . This matches, up to polylogarithmic factors, the
fastest failure-free distance oracles with nearly linear space. For single
vertex failures (), our -size,
-query-time oracle improves over the previously best
known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for , . For multiple failures, no planarity exploiting
results were previously known
Better Tradeoffs for Exact Distance Oracles in Planar Graphs
We present an -space distance oracle for directed planar graphs
that answers distance queries in time. Our oracle both
significantly simplifies and significantly improves the recent oracle of
Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses
-space and answers queries in time. We achieve this by
designing an elegant and efficient point location data structure for Voronoi
diagrams on planar graphs.
We further show a smooth tradeoff between space and query-time. For any , we show an oracle of size that answers queries in time. This new tradeoff is currently the best (up to
polylogarithmic factors) for the entire range of and improves by polynomial
factors over all the previously known tradeoffs for the range
Efficient Wireless Security Through Jamming, Coding and Routing
There is a rich recent literature on how to assist secure communication
between a single transmitter and receiver at the physical layer of wireless
networks through techniques such as cooperative jamming. In this paper, we
consider how these single-hop physical layer security techniques can be
extended to multi-hop wireless networks and show how to augment physical layer
security techniques with higher layer network mechanisms such as coding and
routing. Specifically, we consider the secure minimum energy routing problem,
in which the objective is to compute a minimum energy path between two network
nodes subject to constraints on the end-to-end communication secrecy and
goodput over the path. This problem is formulated as a constrained optimization
of transmission power and link selection, which is proved to be NP-hard.
Nevertheless, we show that efficient algorithms exist to compute both exact and
approximate solutions for the problem. In particular, we develop an exact
solution of pseudo-polynomial complexity, as well as an epsilon-optimal
approximation of polynomial complexity. Simulation results are also provided to
show the utility of our algorithms and quantify their energy savings compared
to a combination of (standard) security-agnostic minimum energy routing and
physical layer security. In the simulated scenarios, we observe that, by
jointly optimizing link selection at the network layer and cooperative jamming
at the physical layer, our algorithms reduce the network energy consumption by
half
Massively Parallel Algorithms for Distance Approximation and Spanners
Over the past decade, there has been increasing interest in
distributed/parallel algorithms for processing large-scale graphs. By now, we
have quite fast algorithms -- usually sublogarithmic-time and often
-time, or even faster -- for a number of fundamental graph
problems in the massively parallel computation (MPC) model. This model is a
widely-adopted theoretical abstraction of MapReduce style settings, where a
number of machines communicate in an all-to-all manner to process large-scale
data. Contributing to this line of work on MPC graph algorithms, we present
round MPC algorithms for computing
-spanners in the strongly sublinear regime of local memory. To
the best of our knowledge, these are the first sublogarithmic-time MPC
algorithms for spanner construction. As primary applications of our spanners,
we get two important implications, as follows:
-For the MPC setting, we get an -round algorithm for
approximation of all pairs shortest paths (APSP) in the
near-linear regime of local memory. To the best of our knowledge, this is the
first sublogarithmic-time MPC algorithm for distance approximations.
-Our result above also extends to the Congested Clique model of distributed
computing, with the same round complexity and approximation guarantee. This
gives the first sub-logarithmic algorithm for approximating APSP in weighted
graphs in the Congested Clique model
- …