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26
On the locality of distributed sparse spanner construction
- In ACM Press, editor, 27th Annual ACM Symp. on Principles of Distributed Computing (PODC
, 2008
"... The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k)edgesforeveryn-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O ..."
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The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k)edgesforeveryn-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O(kn 1+1/k) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ɛ>0, constructs a (1 + ɛ, 2)-spanner of O(ɛ −1 n 3/2)edgesin O(ɛ −1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k − 1, 0)-spanner of o(n 1+1/(k−1))edgesfork ∈{2, 3, 5}. It is also shown that for every k>1, any (randomized) distributed algorithm that constructs a spanner with fewer than n 1+1/k+ɛ edges in at most n ɛ expected rounds must stretch some distances by an additive factor of n Ω(ɛ).Inotherwords, while additive stretched spanners with O(n 1+1/k) edges may exist, e.g., for k =2, 3, they cannot be computed distributively in a sub-polynomial number of rounds in expectation. Supported by the équipe-projet INRIA “DOLPHIN”. Supported by the ANR-project “ALADDIN”, and the
Fault-tolerant spanners for general graphs
- in STOC’09, 2009
"... The paper concerns graph spanners that are resistant to ver-tex or edge failures. Given a weighted undirected n-vertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u ..."
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The paper concerns graph spanners that are resistant to ver-tex or edge failures. Given a weighted undirected n-vertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u, v) denotes the distance between u and v in G Graph spanners were extensively studied since their intro-duction over two decades ago. It is known how to efficiently construct a (2k−1)-spanner of size O(n1+1/k), and this size-stretch tradeoff is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [Levcopoulos et al., STOC’98]. A subgraph H is an f-vertex fault tolerant k-spanner of the graph G if for any set F ⊆ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F (u, v) ≤ k · δG\F (u, v). Levcopoulos et al. presented an efficient algorithm that given a set S of n points in Rd, constructs an f-vertex fault tolerant geometric (1+)-spanner for S, that is, a sparse graph H such that for every set F ⊆ S of size f and any pair of points u, v ∈ S \F, δH\F (u, v) ≤ (1+)|uv|, where |uv | is the Euclidean distance between u and v. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [Czumaj & Zhao, SoCG’03]. This paper also raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an f-vertex fault tolerant (2k−1)-spanner of size
Additive Spanners and (α, β)-Spanners
"... An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). How ..."
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An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1, 6)-spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well-approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)-spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance
, 2012
"... In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each r ..."
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In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ −1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog(n)) rounds in a network of diameter D, with no dependence on the conductance. This is
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
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We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)-spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.
Local Computation of Nearly Additive Spanners
"... An (α, β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β) ..."
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An (α, β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)-spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every n-node graph and integer k ≥ 1, an (α, β)-spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)-spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)-spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)-spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic
Additive spanners in nearly quadratic time
- IN PROCEEDINGS OF THE 37TH INTERNATIONAL COLLOQUIUM CONFERENCE ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP
, 2010
"... We consider the problem of efficiently finding an additive C-spanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u, v, δH(u, v) ≤ δG(u, v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an additive 6-s ..."
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We consider the problem of efficiently finding an additive C-spanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u, v, δH(u, v) ≤ δG(u, v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an additive 6-spanner with O(n 4/3) edges in O(mn 2/3) time. It is unknown if there exists a constant C and an additive C-spanner with o(n 4/3) edges. Moreover, for C ≤ 5 all known constructions require Ω(n 3/2) edges. We give a significantly more efficient construction of an additive 6-spanner. The number of edges in our spanner is n 4/3 polylog n, matching what was previously known up to a polylogarithmic factor, but we greatly improve the time for construction, from O(mn 2/3) to n 2 polylog n. Notice that mn 2/3 ≤ n 2 only if m ≤ n 4/3, but in this case G itself is a sparse spanner. We thus provide both the fastest and the sparsest (up to logarithmic factors) known construction of a spanner with constant additive distortion. We give similar improvements in the construction time of additive spanners under the assumption that the input graph has large girth, or more generally, the input graph has few edges on short cycles.
On approximate distance labels and routing schemes with affine stretch
- IN INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING (DISC
, 2011
"... For every integral parameter k> 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, adistance label L(u) of size Õ(n2/(2k−1)). For any u, v ∈ G, givenL(u),L(v) we can return in time O(k) an affine approximation ˆ d(u, v) on the distance d(u, v) between u and v in ..."
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For every integral parameter k> 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, adistance label L(u) of size Õ(n2/(2k−1)). For any u, v ∈ G, givenL(u),L(v) we can return in time O(k) an affine approximation ˆ d(u, v) on the distance d(u, v) between u and v in G such that d(u, v) � ˆ d(u, v) � (2k − 2)d(u, v) +1. Hence we say that our distance label scheme has affine stretch of (2k − 2)d +1.Fork=2our construction is comparable to the O(n 5/3) size, 2d +1 affine stretch of the distance oracle of Pǎtraşcu and Roditty (FOCS ’10), it incurs a o(log n) storage overhead while providing the benefits of a distance label. For any k>1, givena restriction of o(n 1+1/(k−1) ) on the total size of the data structure, our construction provides distance labels with affine stretch of (2k − 2)d +1 which is better than the stretch (2k − 1)d scheme of Thorup and Zwick (J. ACM ’05). Our second contribution is a compact routing scheme with poly-logarithmic addresses that provides affine stretch guarantees. With Õ(n 3/(3k−2))-bit routing tables we obtain affine stretch of (4k − 6)d +1, for any k>1. Given a restriction of o(n 1/(k−1) ) on the table size, our routing scheme provides affine stretch which is better than the stretch (4k − 5)d routing scheme of Thorup and Zwick (SPAA ’01).
Sparse spanners vs. compact routing.
- In Proc. 23th ACM Symp. on Parallel Algorithms and Architectures (SPAA),
, 2011
"... ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ( ..."
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ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ(
Deterministic Distributed Construction of Linear Stretch Spanners in Polylogarithmic Time
, 2007
"... The paper presents a deterministic distributed algorithm that given an n node unweighted graph constructs an O(n 3/2) edge 3-spanner for it in O(log n) time. This algorithm is then extended into a deterministic algorithm for computing an O(kn 1+1/k) edge O(k)-spanner in O(log k−1 n) time for every i ..."
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The paper presents a deterministic distributed algorithm that given an n node unweighted graph constructs an O(n 3/2) edge 3-spanner for it in O(log n) time. This algorithm is then extended into a deterministic algorithm for computing an O(kn 1+1/k) edge O(k)-spanner in O(log k−1 n) time for every integer parameter k � 1. This establishes that the problem of the deterministic construction of a low stretch spanner with few edges can be solved in the distributed setting in polylogarithmic time. The paper also investigates the distributed construction of sparse spanners with almost pure additive stretch (1 + ɛ, β), i.e., such that the distance in the spanner is at most 1 + ɛ times the original distance plus β. It is shown, for every ɛ> 0, that in O(log n/ɛ) time one can deterministically construct a spanner with O(n 3/2) edges that is both a 3-spanner and a (1 + ɛ, 8 log n)-spanner. Furthermore, it is shown that in n O(1/ √ log n) + O(1/ɛ) time one can deterministically construct a spanner with O(n 3/2) edges which is both a 3-spanner and a (1+ɛ, 4)-spanner. (This algorithm can be transformed into a Las Vegas randomized algorithm with guarantees on the stretch and time, running in O(log n +1/ɛ) expected time).
