21,157 research outputs found
Labeling Schemes with Queries
We study the question of ``how robust are the known lower bounds of labeling
schemes when one increases the number of consulted labels''. Let be a
function on pairs of vertices. An -labeling scheme for a family of graphs
\cF labels the vertices of all graphs in \cF such that for every graph
G\in\cF and every two vertices , the value can be inferred
by merely inspecting the labels of and .
This paper introduces a natural generalization: the notion of -labeling
schemes with queries, in which the value can be inferred by inspecting
not only the labels of and but possibly the labels of some additional
vertices. We show that inspecting the label of a single additional vertex (one
{\em query}) enables us to reduce the label size of many labeling schemes
significantly
On Compact Routing for the Internet
While there exist compact routing schemes designed for grids, trees, and
Internet-like topologies that offer routing tables of sizes that scale
logarithmically with the network size, we demonstrate in this paper that in
view of recent results in compact routing research, such logarithmic scaling on
Internet-like topologies is fundamentally impossible in the presence of
topology dynamics or topology-independent (flat) addressing. We use analytic
arguments to show that the number of routing control messages per topology
change cannot scale better than linearly on Internet-like topologies. We also
employ simulations to confirm that logarithmic routing table size scaling gets
broken by topology-independent addressing, a cornerstone of popular
locator-identifier split proposals aiming at improving routing scaling in the
presence of network topology dynamics or host mobility. These pessimistic
findings lead us to the conclusion that a fundamental re-examination of
assumptions behind routing models and abstractions is needed in order to find a
routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802
Compact Routing on Internet-Like Graphs
The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing
scheme delivering a nearly optimal local memory upper bound. Using both direct
analysis and simulation, we calculate the stretch distribution of this routing
scheme on random graphs with power-law node degree distributions, . We find that the average stretch is very low and virtually
independent of . In particular, for the Internet interdomain graph,
, the average stretch is around 1.1, with up to 70% of paths
being shortest. As the network grows, the average stretch slowly decreases. The
routing table is very small, too. It is well below its upper bounds, and its
size is around 50 records for -node networks. Furthermore, we find that
both the average shortest path length (i.e. distance) and width of
the distance distribution observed in the real Internet inter-AS graph
have values that are very close to the minimums of the average stretch in the
- and -directions. This leads us to the discovery of a unique
critical quasi-stationary point of the average TZ stretch as a function of
and . The Internet distance distribution is located in a
close neighborhood of this point. This observation suggests the analytical
structure of the average stretch function may be an indirect indicator of some
hidden optimization criteria influencing the Internet's interdomain topology
evolution.Comment: 29 pages, 16 figure
Dynamic and Multi-functional Labeling Schemes
We investigate labeling schemes supporting adjacency, ancestry, sibling, and
connectivity queries in forests. In the course of more than 20 years, the
existence of labeling schemes supporting each of these
functions was proven, with the most recent being ancestry [Fraigniaud and
Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower
or upper bounds of or
respectively. Notably an upper bound of for
adjacency+siblings and a lower bound of for each of the
functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We
improve the constants hidden in the -notation. In particular we show a lower bound for connectivity+ancestry and
connectivity+siblings, as well as an upper bound of for connectivity+adjacency+siblings by altering existing
methods.
In the context of dynamic labeling schemes it is known that ancestry requires
bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower
bounds on the label size for adjacency, siblings, and connectivity of
bits, and to support all three functions. There exist efficient
adjacency labeling schemes for planar, bounded treewidth, bounded arboricity
and interval graphs. In a dynamic setting, we show a lower bound of
for each of those families.Comment: 17 pages, 5 figure
A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs
The Capacitated Vehicle Routing problem is to find a minimum-cost set of
tours that collectively cover clients in a graph, such that each tour starts
and ends at a specified depot and is subject to a capacity bound on the number
of clients it can serve. In this paper, we present a polynomial-time
approximation scheme (PTAS) for instances in which the input graph is planar
and the capacity is bounded. Previously, only a quasipolynomial-time
approximation scheme was known for these instances. To obtain this result, we
show how to embed planar graphs into bounded-treewidth graphs while preserving,
in expectation, the client-to-client distances up to a small additive error
proportional to client distances to the depot
Near-optimal adjacency labeling scheme for power-law graphs
An adjacency labeling scheme is a method that assigns labels to the vertices
of a graph such that adjacency between vertices can be inferred directly from
the assigned label, without using a centralized data structure. We devise
adjacency labeling schemes for the family of power-law graphs. This family that
has been used to model many types of networks, e.g. the Internet AS-level
graph. Furthermore, we prove an almost matching lower bound for this family. We
also provide an asymptotically near- optimal labeling scheme for sparse graphs.
Finally, we validate the efficiency of our labeling scheme by an experimental
evaluation using both synthetic data and real-world networks of up to hundreds
of thousands of vertices
Distance labeling schemes for trees
We consider distance labeling schemes for trees: given a tree with nodes,
label the nodes with binary strings such that, given the labels of any two
nodes, one can determine, by looking only at the labels, the distance in the
tree between the two nodes.
A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg
(J. Graph Theory 2000) establish that labels must use
bits\footnote{Throughout this paper we use for .}. Gavoille et.
al. (ESA 2001) show that for very small approximate stretch, labels use
bits. Several other papers investigate various
variants such as, for example, small distances in trees (Alstrup et. al.,
SODA'03).
We improve the known upper and lower bounds of exact distance labeling by
showing that bits are needed and that bits are sufficient. We also give ()-stretch labeling
schemes using bits for constant .
()-stretch labeling schemes with polylogarithmic label size have
previously been established for doubling dimension graphs by Talwar (STOC
2004).
In addition, we present matching upper and lower bounds for distance labeling
for caterpillars, showing that labels must have size . For simple paths with nodes and edge weights in , we show that
labels must have size
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
- …