1,081 research outputs found
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
Vertex Fault-Tolerant Emulators
A -spanner of a graph is a sparse subgraph that preserves its shortest
path distances up to a multiplicative stretch factor of , and a -emulator
is similar but not required to be a subgraph of . A classic theorem by
Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available
to emulators, the size/stretch tradeoffs for spanners and emulators are
equivalent. Our main result is that this equivalence in tradeoffs no longer
holds in the commonly-studied setting of graphs with vertex failures. That is:
we introduce a natural definition of vertex fault-tolerant emulators, and then
we show a three-way tradeoff between size, stretch, and fault-tolerance for
these emulators that polynomially surpasses the tradeoff known to be optimal
for spanners.
We complement our emulator upper bound with a lower bound construction that
is essentially tight (within factors of the upper bound) when the
stretch is and is either a fixed odd integer or . We also show
constructions of fault-tolerant emulators with additive error, demonstrating
that these also enjoy significantly improved tradeoffs over those available for
fault-tolerant additive spanners.Comment: To appear in ITCS 202
Approximation Algorithms and Hardness for -Pairs Shortest Paths and All-Nodes Shortest Cycles
We study the approximability of two related problems on graphs with nodes
and edges: -Pairs Shortest Paths (-PSP), where the goal is to find a
shortest path between prespecified pairs, and All Node Shortest Cycles
(ANSC), where the goal is to find the shortest cycle passing through each node.
Approximate -PSP has been previously studied, mostly in the context of
distance oracles. We ask the question of whether approximate -PSP can be
solved faster than by using distance oracles or All Pair Shortest Paths (APSP).
ANSC has also been studied previously, but only in terms of exact algorithms,
rather than approximation. We provide a thorough study of the approximability
of -PSP and ANSC, providing a wide array of algorithms and conditional lower
bounds that trade off between running time and approximation ratio.
A highlight of our conditional lower bounds results is that for any integer
, under the combinatorial -clique hypothesis, there is no
combinatorial algorithm for unweighted undirected -PSP with approximation
ratio better than that runs in
time. This nearly matches an upper bound implied by the result of Agarwal
(2014).
A highlight of our algorithmic results is that one can solve both -PSP and
ANSC in time with approximation factor
(and additive error that is function of ), for any
constant . For -PSP, our conditional lower bounds imply that
this approximation ratio is nearly optimal for any subquadratic-time
combinatorial algorithm. We further extend these algorithms for -PSP and
ANSC to obtain a time/accuracy trade-off that includes near-linear time
algorithms.Comment: Abstract truncated to meet arXiv requirement. To appear in FOCS 202
Fault-Tolerant Spanners against Bounded-Degree Edge Failures: Linearly More Faults, Almost For Free
We study a new and stronger notion of fault-tolerant graph structures whose
size bounds depend on the degree of the failing edge set, rather than the total
number of faults. For a subset of faulty edges , the
faulty-degree is the largest number of faults in incident to any
given vertex. We design new fault-tolerant structures with size comparable to
previous constructions, but which tolerate every fault set of small
faulty-degree , rather than only fault sets of small size . Our
main results are:
- New FT-Certificates: For every -vertex graph and degree threshold
, one can compute a connectivity certificate with edges that has the following guarantee: for any edge set
with faulty-degree and every vertex pair , it holds that
and are connected in iff they are connected in . This bound on is nearly tight. Since our certificates
handle some fault sets of size up to , prior work did not imply any
nontrivial upper bound for this problem, even when .
- New FT-Spanners: We show that every -vertex graph admits a
-spanner with edges, which
tolerates any fault set of faulty-degree at most . This bound on
optimal up to its hidden dependence on , and it is close to the
bound of that is known for the case where the
total number of faults is [Bodwin, Dinitz, Robelle SODA '22]. Our proof
of this theorem is non-constructive, but by following a proof strategy of
Dinitz and Robelle [PODC '20], we show that the runtime can be made polynomial
by paying an additional factor in spanner size
Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free
A t-emulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant t-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior.
In particular, our main result is that, for (2k-1)-emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any n-node input graph, we construct a 5-emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that ?(n^{4/3}) edges are necessary even if the 5-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k
Adaptive fault-tolerant routing in hypercube multicomputers
A connected hypercube with faulty links and/or nodes is called an injured hypercube. To enable any non-faulty node to communicate with any other non-faulty node in an injured hypercube, the information on component failures has to be made available to non-faulty nodes so as to route messages around the faulty components. A distributed adaptive fault tolerant routing scheme is proposed for an injured hypercube in which each node is required to know only the condition of its own links. Despite its simplicity, this scheme is shown to be capable of routing messages successfully in an injured hypercube as long as the number of faulty components is less than n. Moreover, it is proved that this scheme routes messages via shortest paths with a rather high probabiltiy and the expected length of a resulting path is very close to that of a shortest path. Since the assumption that the number of faulty components is less than n in an n-dimensional hypercube might limit the usefulness of the above scheme, a routing scheme is introduced based on depth-first search which works in the presence of an arbitrary number of faulty components. Due to the insufficient information on faulty components, the paths chosen by the above scheme may not always be the shortest. To guarantee that all messages be routed via shortest paths, it is proposed that every mode be equipped with more information than that on its own links. The effects of this additional information on routing efficiency are analyzed, and the additional information to be kept at each node for the shortest path routing is determined. Several examples and remarks are also given to illustrate the results
An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model
In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph G=(V,E) with n=|V| and m=|E|, and an integer value kgeq 1, there is an algorithm that computes in O(2^{k}n log^2 n) time for any set F of size at most k the strongly connected components of the graph GF. The running time of our algorithm is almost optimal since the time for outputting the SCCs of GF is at least Omega(n). The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size O(2^{k} n^2).
Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected components that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. This new observation alone, however, is not enough, since we need to find an efficient way to represent the strongly connected components using paths. For this purpose we use a mixture of old and classical techniques such as the heavy path decomposition of Sleator and Tarjan and the classical Depth-First-Search algorithm. Although, these are by now standard techniques, we are not aware of any usage of them in the context of dynamic maintenance of SCCs. Therefore, we expect that our new insights and mixture of new and old techniques will be of independent interest
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
On Fast and Robust Information Spreading in the Vertex-Congest Model
This paper initiates the study of the impact of failures on the fundamental
problem of \emph{information spreading} in the Vertex-Congest model, in which
in every round, each of the nodes sends the same -bit message
to all of its neighbors.
Our contribution to coping with failures is twofold. First, we prove that the
randomized algorithm which chooses uniformly at random the next message to
forward is slow, requiring rounds on some graphs, which we
denote by , where is the vertex-connectivity.
Second, we design a randomized algorithm that makes dynamic message choices,
with probabilities that change over the execution. We prove that for
it requires only a near-optimal number of rounds, despite a
rate of failures per round. Our technique of choosing
probabilities that change according to the execution is of independent
interest.Comment: Appears in SIROCCO 2015 conferenc
Extraconnectivity of k-ary n-cube networks
AbstractGiven a graph G and a non-negative integer g, the g-extraconnectivity of G is the minimum cardinality of a set of vertices in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with more than g vertices. This study shows that the 2-extraconnectivity of a k-ary n-cube Qnk for k≥4 and n≥5 is equal to 6n−5
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