2,159 research outputs found
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Space Efficient Edge-Fault Tolerant Routing
Let G be an undirected weighted graph with n vertices and m edges, and k >= 1 be an integer. We preprocess the graph in O^~(mn) time, constructing a data structure of size O^~ k deg{v}+n^{1/k}) words per vertex v in V, which is then used by our routing scheme to ensure successful routing of packets even in the presence of a single edge fault. The scheme adds only O(k) words of information to the message.
Moreover, the stretch of the routing scheme, i.e., the maximum ratio of the cost of the path along which the packet is routed to the cost of the actual shortest path that avoids the fault, is only O(k^2).
Our results match the best known results for routing schemes that do not consider failures, with only the stretch being larger by a small constant factor of O(k). Moreover, a 1963 girth conjecture of Erdos, known to hold for k=1,2,3 and 5, implies that Omega(n^{1+1/k}) space is required by any routing scheme that has a stretch less than 2k+1.
Hence our data structures are essentially space efficient.
The algorithms are extremely simple, easy to implement, and with minor modifications, can be used under a centralized setting to efficiently answer distance queries in the presence of faults.
An important component of our routing scheme that may be of independent interest is an algorithm to compute the shortest cycle passing through each edge. As an intermediate result, we show that computing this in a distributed model that stores at each vertex the shortest path tree rooted at that node requires Theta(mn) message passings in the worst case
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
Stochastic Analysis of a Churn-Tolerant Structured Peer-to-Peer Scheme
We present and analyze a simple and general scheme to build a churn
(fault)-tolerant structured Peer-to-Peer (P2P) network. Our scheme shows how to
"convert" a static network into a dynamic distributed hash table(DHT)-based P2P
network such that all the good properties of the static network are guaranteed
with high probability (w.h.p). Applying our scheme to a cube-connected cycles
network, for example, yields a degree connected network, in which
every search succeeds in hops w.h.p., using messages,
where is the expected stable network size. Our scheme has an constant
storage overhead (the number of nodes responsible for servicing a data item)
and an overhead (messages and time) per insertion and essentially
no overhead for deletions. All these bounds are essentially optimal. While DHT
schemes with similar guarantees are already known in the literature, this work
is new in the following aspects:
(1) It presents a rigorous mathematical analysis of the scheme under a
general stochastic model of churn and shows the above guarantees;
(2) The theoretical analysis is complemented by a simulation-based analysis
that validates the asymptotic bounds even in moderately sized networks and also
studies performance under changing stable network size;
(3) The presented scheme seems especially suitable for maintaining dynamic
structures under churn efficiently. In particular, we show that a spanning tree
of low diameter can be efficiently maintained in constant time and logarithmic
number of messages per insertion or deletion w.h.p.
Keywords: P2P Network, DHT Scheme, Churn, Dynamic Spanning Tree, Stochastic
Analysis
Robust Routing Made Easy
Designing routing schemes is a multidimensional and complex task that depends on the objective function, the computational model (centralized vs. distributed), and the amount of uncertainty (online vs. offline). Nevertheless, there are quite a few well-studied general techniques, for a large variety of network problems. In contrast, in our view, practical techniques for designing robust routing schemes are scarce; while fault-tolerance has been studied from a number of angles, existing approaches are concerned with dealing with faults after the fact by rerouting, self-healing, or similar techniques. We argue that this comes at a high burden for the designer, as in such a system any algorithm must account for the effects of faults on communication. With the goal of initiating efforts towards addressing this issue, we showcase simple and generic transformations that can be used as a blackbox to increase resilience against (independently distributed) faults. Given a network and a routing scheme, we determine a reinforced network and corresponding routing scheme that faithfully preserves the specification and behavior of the original scheme. We show that reasonably small constant overheads in terms of size of the new network compared to the old are sufficient for substantially relaxing the reliability requirements on individual components. The main message in this paper is that the task of designing a robust routing scheme can be decoupled into (i) designing a routing scheme that meets the specification in a fault-free environment, (ii) ensuring that nodes correspond to fault-containment regions, i.e., fail (approximately) independently, and (iii) applying our transformation to obtain a reinforced network and a robust routing scheme that is fault-tolerant
Compact routing in fault-tolerant distributed systems
A compact routing algorithm is a routing algorithm which reduces the space complexity of all-pairs shortest path routing. Compact routing protocols in distributed systems have been studied extensively as an attractive alternative to the traditional method of all-pairs shortest path routing. The use of compact routing protocols have several advantages. Compact routing schemes are not only more memory-efficient, but provide faster routing table lookup, more efficient broadcast scheme, and allow for a more scalable network. These routing schemes still maintain optimal or near-optimal routing paths. However, most of the compact routing protocols are not fault-tolerant. This thesis will first report the recent developments in the compact routing research. Several new methods for compact routing in fault-tolerant distributed systems will be presented and analyzed. The most important feature of the algorithms presented in this thesis is that they are self-stabilizing. The self-stabilization paradigm has been shown to be the most unified and all-inclusive approach to the design of fault-tolerant system. Additionally, these algorithms will address and solve several problems left unsolved by previous works. Relabelable and non-relabelable networks will be considered for both specific and arbitrary topologies
- …