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Low Distortion Spanners
"... A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tr ..."
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A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood. In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. and Baswana et al. and improve on the (1+ɛ, β)-spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially.
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.
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 Pairwise Spanners
, 2013
"... Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparses ..."
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Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the u-v distance only for pairs (u, v) in a given set P ⊆ V × V. Such P-spanners were studied before [Coppersmith,Elkin’05] only in the special case that f(·) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(·)). In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P | log n) 1/4) with f(d) = d+4 log n. Alternatively, for any ε> 0, there exists a P-spanner of size O(n|P | 1/4 log n ε) with f(d) = (1 + ε)d + 4. We also consider the relevant special case that there is a critical set of nodes S ⊆ V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S × S)-spanner of size O(n √ |S|) with f(d) = d + 2, and an (S × V)-spanner of size O(n √ |S | log n) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.
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(
Fault tolerant approximate bfs structures
- In SODA
, 2014
"... A fault-tolerant structure for a network is required to continue functioning fol-lowing the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a fault-tolerant (α, β) approximate BFS structure (or FT-ABFS structure for short), namely, a subgraph H of th ..."
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A fault-tolerant structure for a network is required to continue functioning fol-lowing the failure of some of the network’s edges or vertices. This paper addresses the problem of designing a fault-tolerant (α, β) approximate BFS structure (or FT-ABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s, v,H \ F) ≤ α · dist(s, v,G \ F) + β for every v ∈ V. We first consider multiplicative (α, 0) FT-ABFS structures resilient to a failure of a single edge and present an algorithm that given an n-vertex unweighted undi-rected graph G and a source s constructs a (3, 0) FT-ABFS structure rooted at s with at most 4n edges (improving by an O(log n) factor on the near-tight result of [3] for the special case of edge failures). Assuming at most f edge failures, for constant integer f> 1, we prove that there exists a (poly-time constructible) (3(f + 1), (f + 1) log n) FT-ABFS structure with O(fn) edges. We then consider additive (1, β) FT-ABFS structures. In contrast to the linear size of (α, 0) FT-ABFS structures, we show that for every β ∈ [1, O(log n)] there exists an n-vertex graph G with a source s for which any (1, β) FT-ABFS structure rooted at s has Ω(n1+(β)) edges, for some function (β) ∈ (0, 1). In particular, (1, 3) FT-ABFS structures admit a lower bound of Ω(n5/4) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive
Efficient Distributed Source Detection with Limited Bandwidth
"... Given a simple graph G = (V, E) and a set of sources S ⊆ V, denote for each node v ∈ V by L (∞) v 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 ..."
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Given a simple graph G = (V, E) and a set of sources S ⊆ V, denote for each node v ∈ V by L (∞) v 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 (if for all of them d(s, v) ≤ d) or all entries (d(s, v), s) ∈ L (∞) v 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. first k entries of L (∞)
Vertex Fault Tolerant Additive Spanners
, 2014
"... A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a fault-tolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failu ..."
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A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a fault-tolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H still contains an additive spanner for (the surviving part of) G, satisfying dist(s, t,H \ {v}) ≤ dist(s, t,G \ {v}) + β for every s, t, v ∈ V. Recently, the problem of constructing fault-tolerant additive spanners resilient to the failure of up to f edges has been considered [8]. The problem of handling vertex failures was left open therein. In this paper we develop new techniques for constructing additive FT-spanners overcoming the failure of a single vertex in the graph. Our first result is an FT-spanner with additive stretch 2 and Õ(n5/3) edges. Our second result is an FT-spanner with additive stretch 6 and Õ(n3/2) edges. The construction algorithm consists of two main components: (a) constructing an FT-clustering graph and (b) applying a modified path-buying procedure suitably adopted to failure prone settings. Finally, we also describe two constructions for fault-tolerant multi-source additive spanners, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S×V for a given subset of sources S ⊆ V. The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges).
