7,236 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
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
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
Fast Routing Table Construction Using Small Messages
We describe a distributed randomized algorithm computing approximate
distances and routes that approximate shortest paths. Let n denote the number
of nodes in the graph, and let HD denote the hop diameter of the graph, i.e.,
the diameter of the graph when all edges are considered to have unit weight.
Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD)
communication rounds using messages of O(log n) bits and guarantees a stretch
of O(eps^(-1) log eps^(-1)) with high probability. This is the first
distributed algorithm approximating weighted shortest paths that uses small
messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time
complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the
small-messages model that hold for stateless routing (where routing decisions
do not depend on the traversed path) as well as approximation of the weigthed
diameter. Our scheme replaces the original identifiers of the nodes by labels
of size O(log eps^(-1) log n). We show that no algorithm that keeps the
original identifiers and runs for weak-o(n) rounds can achieve a
polylogarithmic approximation ratio.
Variations of our techniques yield a number of fast distributed approximation
algorithms solving related problems using small messages. Specifically, we
present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0
< eps <= 1/2, and solve, with high probability, the following problems:
- O(eps^(-1))-approximation for the Generalized Steiner Forest (the running
time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the
number of terminals);
- O(eps^(-2))-approximation of weighted distances, using node labels of size
O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node;
- O(eps^(-1))-approximation of the weighted diameter;
- O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1
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
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
On space-stretch trade-offs: upper bounds
One of the fundamental trade-offs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the maximum ratio over all pairs between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. All previous routing schemes required storage that is dependent on the diameter of the network. We present a new scale-free routing scheme, whose storage and header sizes are independent of the aspect ratio of the network. Our scheme is based on a decomposition into sparse and dense neighborhoods. Given an undirected network with arbitrary weights and n arbitrary node names, for any integer k ≥ 1 we present the first scale-free routing scheme with asymptotically optimal space-stretch tradeoff that does not require edge weights to be polynomially bounded. The scheme uses e O(n 1/k) space routing table at each node, and routes along paths of asymptotically optimal linear stretch O(k)
On the Memory Requirement of Hop-by-hop Routing: Tight Bounds and Optimal Address Spaces
Routing in large-scale computer networks today is built on hop-by-hop routing: packet headers specify the destination address and routers use internal forwarding tables to map addresses to next-hop ports. In this paper we take a new
look at the scalability of this paradigm. We define a new model that reduces forwarding tables to sequential strings, which then lend themselves readily to an information-theoretical analysis. Contrary to previous work, our analysis is not of worst-case nature, but gives verifiable and realizable memory requirement characterizations even when subjected to concrete topologies and routing policies. We formulate the optimal address space design problem as the task to set node addresses in order to minimize certain network-wide entropy-related measures. We derive tight space bounds for many well-known graph families and we propose a simple heuristic to find optimal address spaces for general graphs. Our evaluations suggest that in structured graphs, including most practically important network topologies,
significant memory savings can be attained by forwarding table compression over our optimized address spaces. According to our knowledge, our work is the first to bridge the gap between computer network scalability and information-theory
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