6,777 research outputs found
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
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
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
Compact Oblivious Routing
Oblivious routing is an attractive paradigm for large distributed systems in which centralized control and frequent reconfigurations are infeasible or undesired (e.g., costly). Over the last almost 20 years, much progress has been made in devising oblivious routing schemes that guarantee close to optimal load and also algorithms for constructing such schemes efficiently have been designed. However, a common drawback of existing oblivious routing schemes is that they are not compact: they require large routing tables (of polynomial size), which does not scale.
This paper presents the first oblivious routing scheme which guarantees close to optimal load and is compact at the same time - requiring routing tables of polylogarithmic size. Our algorithm maintains the polylogarithmic competitive ratio of existing algorithms, and is hence particularly well-suited for emerging large-scale networks
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
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
On Efficient Distributed Construction of Near Optimal Routing Schemes
Given a distributed network represented by a weighted undirected graph
on vertices, and a parameter , we devise a distributed
algorithm that computes a routing scheme in
rounds, where is the hop-diameter of the network. The running time matches
the lower bound of rounds (which holds for any
scheme with polynomial stretch), up to lower order terms. The routing tables
are of size , the labels are of size , and
every packet is routed on a path suffering stretch at most . Our
construction nearly matches the state-of-the-art for routing schemes built in a
centralized sequential manner. The previous best algorithms for building
routing tables in a distributed small messages model were by \cite[STOC
2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but
suffers from substantially larger routing tables of size ,
while the latter has sub-optimal running time of
Near-optimal labeling schemes for nearest common ancestors
We consider NCA labeling schemes: given a rooted tree , label the nodes of
with binary strings such that, given the labels of any two nodes, one can
determine, by looking only at the labels, the label of their nearest common
ancestor.
For trees with nodes we present upper and lower bounds establishing that
labels of size , are both sufficient and
necessary. (All logarithms in this paper are in base 2.)
Alstrup, Bille, and Rauhe (SIDMA'05) showed that ancestor and NCA labeling
schemes have labels of size . Our lower bound
increases this to for NCA labeling schemes. Since
Fraigniaud and Korman (STOC'10) established that labels in ancestor labeling
schemes have size , our new lower bound separates
ancestor and NCA labeling schemes. Our upper bound improves the
upper bound by Alstrup, Gavoille, Kaplan and Rauhe (TOCS'04), and our
theoretical result even outperforms some recent experimental studies by Fischer
(ESA'09) where variants of the same NCA labeling scheme are shown to all have
labels of size approximately
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