The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph @math and a spanning tree @math for it, and the goal is to augment @math with a minimum set of edges @math from @math , such that @math is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and J ' a J ' a , SICOMP 1981. Recently, a 3 2-approximation was given for the unweighted case by Kortsarz and Nutov, TALG 2016, and recent breakthroughs by Adjiashvili, SODA 2017, and by , 2017, give approximations better than 2 for bounded weights. In this paper, we provide the first fast distributed approximations for TAP. We present a distributed @math -approximation for weighted TAP which completes in @math rounds, where @math is the height of @math . When @math is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in @math rounds, where @math is the number of vertices and @math is the diameter of @math . Immediate consequences of our results are an @math -round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an @math -round 3-approximation algorithm for the weighted case, where @math is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP.
Another related problem is the construction of fault-tolerant tree structures. Distributed algorithms for constructing fault tolerant BFS and MST structures are given in @cite_5 , producing sparse subgraphs of the input graph @math that contain a BFS (or an MST) of @math for each edge @math , for the purpose of maintaining the functionality of a BFS (or an MST) even when an edge fails. However, TAP is different from these problems in several aspects. First, we augment a specific spanning tree @math rather then build the whole structure from scratch. In addition, since we need to preserve only connectivity when an edge fails and not the functionality of a BFS or an MST, optimal solutions for TAP may be much cheaper.
@cite_5
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Tree structures such as breadth-first search (BFS) trees and minimum spanning trees (MST) are among the most fundamental graph structures in distributed network algorithms. However, by definition, these structures are not robust against failures and even a single edge's removal can disrupt their functionality. A well-studied concept which attempts to circumvent this issue is Fault-Tolerant Tree Structures, where the tree gets augmented with additional edges from the network so that the functionality of the structure is maintained even when an edge fails. These structures, or other equivalent formulations, have been studied extensively from a centralized viewpoint. However, despite the fact that the main motivations come from distributed networks, their distributed construction has not been addressed before. In this paper, we present distributed algorithms for constructing fault tolerant BFS and MST structures. The time complexity of our algorithms are nearly optimal in the following strong sense: they almost match even the lower bounds of constructing (basic) BFS and MST trees. A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults @math , for all stretch bounds. For stretch @math we design a simple transformation that converts every @math -spanner construction with at most @math edges into an @math -fault-tolerant @math -spanner construction with at most @math edges. Applying this to standard greedy spanner constructions gives @math -fault tolerant @math -spanners with @math edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on @math but exponentially on @math (approximately like @math ). For the case @math and unit-length edges, an @math -approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to @math , which is, notably, independent of the number of faults @math . We further strengthen this bound in terms of the maximum degree by using the Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation. This paper studies the problem of constructing a minimum-weight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today is a protocol with running time O(Λ(G)+n⋅log∗n) due to Kutten and Peleg [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27], where Λ(G) denotes the diameter of the graph G. Peleg and Rubinovich [D. Peleg, V. Rubinovich, A near-tight lower bound on the time complexity of distributed MST construction, in: Proc. 40th IEEE Symp. on Foundations of Computer Science, 1999, pp. 253–261] have shown that Ω˜(n) time is required for constructing MST even on graphs of small diameter, and claimed that their result “establishes the asymptotic near-optimality” of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. In this paper we refine this claim, and devise a protocol that constructs the MST in Ω˜(μ(G,ω)+n) rounds, where μ(G,ω) is the MST-radius of the graph. The ratio between the diameter and the MST-radius may be as large as Θ(n), and, consequently, on some inputs our protocol is faster than the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27] by a factor of Ω˜(n). Also, on every input, the running time of our protocol is never greater than twice the running time of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. As part of our protocol for constructing an MST, we develop a protocol for constructing neighborhood covers with a drastically improved running time. The latter result may be of independent interest. Given a simple graph G=(V,E) and a set of sources S ⊆ V, denote for each node ν e V by Lν(∞) the lexicographically ordered list of distance source pairs (d(s,v),s), where s ∈ S. For integers d,k ∈ N∪ ∞ , we consider the source detection, or (S,d,k)-detection task, requiring each node v to learn the first k entries of Lν(∞) (if for all of them d(s,v) ≤ d) or all entries (d(s,v),s) ∈ Lν(∞) satisfying that d(s,v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k=∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d=∞ and S=V requires constructing a partial BFS tree comprising at least k nodes from every node in V. In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d+k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions.