3,111 research outputs found
Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs
A greedy embedding of a graph into a metric space is a
function such that in the embedding for every pair of
non-adjacent vertices there exists another vertex adjacent
to which is closer to than . This notion of greedy
embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci.
2005), where authors conjectured that every 3-connected planar graph has a
greedy embedding (possibly planar and convex) in the Euclidean plane. Recently,
greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008).
However, their algorithm do not result in a drawing that is planar and convex
for all 3-connected planar graph in the Euclidean plane. In this work we
consider the planar convex greedy embedding conjecture and make some progress.
We derive a new characterization of planar convex greedy embedding that given a
3-connected planar graph , an embedding x: V \to \bbbr^2 of is
a planar convex greedy embedding if and only if, in the embedding , weight
of the maximum weight spanning tree () and weight of the minimum weight
spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq
(\card{V}-1)^{1 - \delta}, for some .Comment: 19 pages, A short version of this paper has been accepted for
presentation in FCT 2009 - 17th International Symposium on Fundamentals of
Computation Theor
On Leveraging Partial Paths in Partially-Connected Networks
Mobile wireless network research focuses on scenarios at the extremes of the
network connectivity continuum where the probability of all nodes being
connected is either close to unity, assuming connected paths between all nodes
(mobile ad hoc networks), or it is close to zero, assuming no multi-hop paths
exist at all (delay-tolerant networks). In this paper, we argue that a sizable
fraction of networks lies between these extremes and is characterized by the
existence of partial paths, i.e. multi-hop path segments that allow forwarding
data closer to the destination even when no end-to-end path is available. A
fundamental issue in such networks is dealing with disruptions of end-to-end
paths. Under a stochastic model, we compare the performance of the established
end-to-end retransmission (ignoring partial paths), against a forwarding
mechanism that leverages partial paths to forward data closer to the
destination even during disruption periods. Perhaps surprisingly, the
alternative mechanism is not necessarily superior. However, under a stochastic
monotonicity condition between current v.s. future path length, which we
demonstrate to hold in typical network models, we manage to prove superiority
of the alternative mechanism in stochastic dominance terms. We believe that
this study could serve as a foundation to design more efficient data transfer
protocols for partially-connected networks, which could potentially help
reducing the gap between applications that can be supported over disconnected
networks and those requiring full connectivity.Comment: Extended version of paper appearing at IEEE INFOCOM 2009, April
20-25, Rio de Janeiro, Brazi
Remarks on Category-Based Routing in Social Networks
It is well known that individuals can route messages on short paths through
social networks, given only simple information about the target and using only
local knowledge about the topology. Sociologists conjecture that people find
routes greedily by passing the message to an acquaintance that has more in
common with the target than themselves, e.g. if a dentist in Saarbr\"ucken
wants to send a message to a specific lawyer in Munich, he may forward it to
someone who is a lawyer and/or lives in Munich. Modelling this setting,
Eppstein et al. introduced the notion of category-based routing. The goal is to
assign a set of categories to each node of a graph such that greedy routing is
possible. By proving bounds on the number of categories a node has to be in we
can argue about the plausibility of the underlying sociological model. In this
paper we substantially improve the upper bounds introduced by Eppstein et al.
and prove new lower bounds.Comment: 21 page
Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is - and -monotone.
Angle-monotone graphs are -spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half--graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex to any vertex whose
length is within times the Euclidean distance from to .
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
Scalable Routing Easy as PIE: a Practical Isometric Embedding Protocol (Technical Report)
We present PIE, a scalable routing scheme that achieves 100% packet delivery
and low path stretch. It is easy to implement in a distributed fashion and
works well when costs are associated to links. Scalability is achieved by using
virtual coordinates in a space of concise dimensionality, which enables greedy
routing based only on local knowledge. PIE is a general routing scheme, meaning
that it works on any graph. We focus however on the Internet, where routing
scalability is an urgent concern. We show analytically and by using simulation
that the scheme scales extremely well on Internet-like graphs. In addition, its
geometric nature allows it to react efficiently to topological changes or
failures by finding new paths in the network at no cost, yielding better
delivery ratios than standard algorithms. The proposed routing scheme needs an
amount of memory polylogarithmic in the size of the network and requires only
local communication between the nodes. Although each node constructs its
coordinates and routes packets locally, the path stretch remains extremely low,
even lower than for centralized or less scalable state-of-the-art algorithms:
PIE always finds short paths and often enough finds the shortest paths.Comment: This work has been previously published in IEEE ICNP'11. The present
document contains an additional optional mechanism, presented in Section
III-D, to further improve performance by using route asymmetry. It also
contains new simulation result
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